Unbraced Length of Members in a Dome
Unbraced Length of Members in a Dome
(OP)
I have been running analysis of a dome. The dome is 50' in diameter and is approximately 27' in height. The dome is not circular, but instead, is faceted.
I have put the model thru its paces via RISA3D. I have applied the appropriate loads and load cases. Movement of the dome is within tolerable limits under load.
The question I have is with regards to unbraced length of the compression members.
I have attached a sketch of what I am talking about.
The curved ribs are 16" x 8" tubes. The curved ribs are designed as pinned at their base and fixed to the face of the compression ring at the top of dome. Metal deck will be applied to the surface of the dome, thus bracing each curved rib in each rib's weak direction. Of course, the strong dimension of the tube will be unbraced.
I am having trouble getting comfortable with what this unbraced length of tube is. My gut tells me that its the full length of tube, all the way over the top of the dome....but this seems very conservative and not appropriate, as all tubes converge at the top of dome.....in other words, its seems that all tubes would have to fail in compression at the same time along the approximate 27' unbraced length, and not the full unbraced length if assumed to be all the way over the dome.
What do you guys see as the unbraced length of a rib? I am trying to get down to what the allowable compressive stress is in the rib member. I have the flexural stresses and the axial stresses, I just need to best determine the allowable axial stress (I know, I know, ASD 9th) to review the combined stresses of the ribs.
Do you guys know of any good publications regarding this question?
Domes have been around since time.......surely there's some guidance literature out there somewhere?
Thank You for your input!
Marinaman
I have put the model thru its paces via RISA3D. I have applied the appropriate loads and load cases. Movement of the dome is within tolerable limits under load.
The question I have is with regards to unbraced length of the compression members.
I have attached a sketch of what I am talking about.
The curved ribs are 16" x 8" tubes. The curved ribs are designed as pinned at their base and fixed to the face of the compression ring at the top of dome. Metal deck will be applied to the surface of the dome, thus bracing each curved rib in each rib's weak direction. Of course, the strong dimension of the tube will be unbraced.
I am having trouble getting comfortable with what this unbraced length of tube is. My gut tells me that its the full length of tube, all the way over the top of the dome....but this seems very conservative and not appropriate, as all tubes converge at the top of dome.....in other words, its seems that all tubes would have to fail in compression at the same time along the approximate 27' unbraced length, and not the full unbraced length if assumed to be all the way over the dome.
What do you guys see as the unbraced length of a rib? I am trying to get down to what the allowable compressive stress is in the rib member. I have the flexural stresses and the axial stresses, I just need to best determine the allowable axial stress (I know, I know, ASD 9th) to review the combined stresses of the ribs.
Do you guys know of any good publications regarding this question?
Domes have been around since time.......surely there's some guidance literature out there somewhere?
Thank You for your input!
Marinaman






RE: Unbraced Length of Members in a Dome
And yes, there is info out there on domes, some of my textbooks have such, but I'm not in the position to scan them at the moment. But you can probably find some resources online.
RE: Unbraced Length of Members in a Dome
Mike McCann
MMC Engineering
RE: Unbraced Length of Members in a Dome
I hear you guys, but the tension rings only brace the ribs in the weak direction of the tube rib. The strong axis of the tube rib seems to be unbraced fully.
On the flip side, the "tension rings" do not let the rib move toward the interior of the dome nor toward the exterior of the dome.
I'm thinking if I had no "tension rings" at all, the unbraced length of rib would be the full distance along the radius of the dome.....but with "tension rings" the rings do not allow the strong axis of the rib to move outward/toward the interior either.
In other words, the tension rings seem to brace the ribs in both the weak and strong dirctions. Is that what we are saying?
RE: Unbraced Length of Members in a Dome
RE: Unbraced Length of Members in a Dome
I disagree due to the shape of the dome and the angle created at the intersection of the rings - there is vertical bracing there for the strong axis. Look harder and draw some diagrams. You'll see it.
Mike McCann
MMC Engineering
RE: Unbraced Length of Members in a Dome
Note that the column formulas refer only to straight members, any kind of intentional curvature will invalidate them.
RE: Unbraced Length of Members in a Dome
I would tend to use the 27ft as the unbraced length(assuming this is the linear length) for the strong axis.
1. As mentioned the ten/com rings @ 1/4 pts do add some bracing in the stong direction but I would
be hesitant to use them as such because they do not act as a direct brace in the strong direction.
If the loading on the dome was uniform and symmetic(say dead load) then I might look at it but would
be concerned with "snap-thru" and have to evaluate that. Wind, seismic and snow can cause
non-uniform and assymmetric loading on the dome causing doubt in my mind of the inherent stability
that a dome shape renders to it's components.
2. I would not use the full lenth over the dome as these members end as an entity @ the compression
ring at the top. They will have the associated rotation and deflection as a boundary condition @ that
ring.
RE: Unbraced Length of Members in a Dome
Agree with Archie and MSQ. This is like the top half of a bourbon barrel
Also, why not use continuous members for the horizontal braces, ex, right above where you wrote "deck". Is this not a straight line or is there a kink in the X direction there too? If there is a kink, then that joint is unstable.
RE: Unbraced Length of Members in a Dome
The curved vertical element can't buckle outward because of the tension rings as others have noted. They also can't buckle inward as that tension ring is now a compression ring.
Take a look at any trussed tower, and you will definitely find the diagonal elements have been sized for their lengths from node to node! And limit KL/R to 200.
RE: Unbraced Length of Members in a Dome
Some diagonal bracing may be advisable in the planes of the roof to prevent torsional buckling of the overall structure, although cladding could be used for this purpose.
BA
RE: Unbraced Length of Members in a Dome
Regarding the vertical curved members of the dome. If you look at these "column" elements in section, it is a curved column similar to the shape of a bow. This column can only buckle one-way.. .and that is outward. Doesn't the tension ring prevent each vertical rib from expanding outward and this reduce it's unbraced length?
I don't know if I agree that the strong axis is unbraced for the entire 27 'span...
RE: Unbraced Length of Members in a Dome
There are two types of rib in the dome. Type 1 occurs at each corner of the octagon. Type 2 occurs at mid-length of each side. There are eight of each type.
Type 2 has its weak axis braced by a member normal to the rib whereas Type 1 is braced at at angle of 22.5 degrees to the rib in plan. For a Type 2 rib, the major axis is laterally unbraced if the bracing members are pin ended because there is nothing to prevent lateral movement.
At first glance, you may feel that Type 1 ribs are braced because of the 22.5 degree angle each brace makes with the rib but that would be wrong because the angles can all change if each brace is pin connected. Each bracing ring is a mechanism which will not hold its original shape without additional bracing.
The dome must be designed for various load conditions. Under balanced gravity load, the ribs may be more or less equally stressed in compression. Under wind, seismic or unbalanced loading conditions, some ribs may carry more compression than others. Some may even feel a stress reversal.
It is vitally important for the designer to see that the major axis of each rib is unbraced from end to end.
BA
RE: Unbraced Length of Members in a Dome
BA
RE: Unbraced Length of Members in a Dome
I see your point and I agree with it. My earlier posts referrencing the lateral unbraced lenght being 6' to 7' and that bracing in the weak axis was all that was required was based upon the prsumption that the longitudanal ribs were continuous and were shown in segmented pieces in order to model their curvature. That was quite an assumption on my part, I'll admit.
Marinaman, could you comment on that? Are the ribs continous (i.e. with the curvature bent or rolled into them)? Or are they in in segmented pieces with pinned connections? There's a big difference in the mechanics and stability of the two.
RE: Unbraced Length of Members in a Dome
Would you consider the top braced? Or, would it act like a cantilever with KL = 2(27')= 54'.
RE: Unbraced Length of Members in a Dome
RE: Unbraced Length of Members in a Dome
1. if use 27ft as unbraced length in strong direction...what K to use?...I probably would use a K=1 but
would pull the compression ring out with it's different loading and check it as a ring by hand.
2. I would check how the RISA 3D program is treating these rings or was it modeled using segmented
lengths. The same applies to the 16x8 ribs as the theory valid for a curved member may not be the
one RISA is using and may be resulting in unconservative results.
3.JSephan mentioned that the typical AISC formalas do not apply to a curved member.....right now I don't know and would have to research it....
RE: Unbraced Length of Members in a Dome
I think, from what I remember, and I have been in the dome many times, that the largest member is either a 6.75X18 or 24, somewhere in that range. I will have to check when I go there again next summer. You can measure the beams at the concrete intersection in the nosebleed section.
Mike McCann
MMC Engineering
RE: Unbraced Length of Members in a Dome
I am under the impression that the ribs are continuous from deck to upper ring. If segmented, the structure is a mechanism, hence unstable.
The top ring is not braced against rotation about a vertical axis except by roof deck or bracing placed in the plane of the deck. That is true of the other three rings as well. Bracing is needed to prevent torsional failure of the structure. If such bracing is used, KLy is about 7' and KLx is about 27'.
Mike, I cannot comment on the Tacoma Dome as I have not seen it or its design. Perhaps the deck is acting like a spherical shell and the ribs are simply acting as stiffeners.
The structure depicted in this thread has a metal cladding but we have not been told that the cladding is capable of acting structurally as a shell. My earlier comments were based on the adequacy of the members alone with no contribution from the shell. If that assumption is incorrect, the OP should let us know.
BA
RE: Unbraced Length of Members in a Dome
RE: Unbraced Length of Members in a Dome
The ribs are continuous from the spring point to the compression ring at the top. The members just look segmented because I modeled the dome rib members as segments with fixed ends...and when printed out in RISA, the members are graphically shown as each individual member.
The ribs span from the spring line to the compression ring as one continuous piece. Each rib is fully welded to the compression ring, all the way around. The compression ring is a 20" tall, 4" wide tube, fully welded to create one continuous piece. The rib members are pinned at the spring line.
The horizontal members are 14" deep, 6" wide tubes. The horizontal members are fully welded to the curved facet 16x8's, and can be fully welded to the facet corner curved ribs.
The entire dome is fully covered in 1 1/2", type B, galvanized metal deck that will be fastened in a 36/7 pattern (5/8" diameter puddle welds) and sidelap fastened however I want it to be.
What I am not showing in the sketch is that there is another, smaller set of tubes, (10x4's), that act as purlins and span from the spring line to the 14x6 and break down the metal deck span.
I would also like to mention that I reviewed an article in Modern Steel Construction from October 2010 regarding domes. Three domes are shown in this magazine. One is of a similar diameter (50') and similar height, but only uses 10" x 4" rib members, a 1/2" plate as a compression ring, but is also round, not faceted. Seems like small members for what its doing.
Hope this clarifies some of the questions.
I'm trying to get comfortable with the 27' unbraced length......but still not sold.
RE: Unbraced Length of Members in a Dome
RE: Unbraced Length of Members in a Dome
BA
RE: Unbraced Length of Members in a Dome
Regarding the geodesic dome design (I might be missing something here) - How would the arrangement of triangles change the out of plane buckling restraint. I mean if there are ribs continuous from top to bottom then the triangles seem to be providing a similar (although maybe different enough) out of plane restriction to buckling that that the "tension rings" would be.
It seems like the tension rings would provide some buckling restraint. However it is hard to say that they really are braced points as it would depend on how much displacement occurs. I would imagine the structure would need to be "squeezed in" if a rib is to buckle in the perpendicular direction (if that makes any sense). Meaning if one of the vertical members buckles outward then the tension ring would have to displace or pull the sides inward. So is it possible to find what the displacement is based on a 2% outward buckling force applied to the tension ring. Then check to see if adequate stiffness exists?
It seems that when googling "dome roof buckling" (as suggested by IFR's) there are quite a few articles but I couldn't really find anything helpful (atleast not free anyway). It seems as though a common theme was that overall stability evaluated through a non-linear analysis should be done more than it is currently being practiced.
In the end it seems as though the 27' unbraced length it pretty reasonable and may avoid a complex analysis in which other complexities could arise or be missed. Although still a good question that I'd like to get to the bottom of.
A couple of links although I'm not sure they are any help to the question:
http://www.scientific.net/AMR.446-449.121
http://books.google.com/books?id=uREySs8cZoUC&...
EIT
www.HowToEngineer.com
RE: Unbraced Length of Members in a Dome
http://books.google.com/books?id=eTVYJfW_2L8C&...
http://www.domerama.com/wp-content/uploads/2012/08...
http://web.mscsoftware.com/support/library/conf/wu...
http://www.domerama.com/calculators/geodesic-analy...
RE: Unbraced Length of Members in a Dome
The side length of the octagon is about 19' at the spring line, so the 10x4 tube purlins could be spaced at third points, providing a maximum span of about 6'-5" for the deck.
BA
RE: Unbraced Length of Members in a Dome
Well, if that's the case then I will stick by my original opinion that the unbraced length is on the order of 6' to 7'. That's presuming the bracing elements have some capacity to them and engage the ribs' compression side. But if this is wrong I'm willing to be taught otherwise, as long as the how's and wherefore's are explained to me.
Also, Marinaman, since the ribs are modeled as segments make sure that the end-to-end connenctions are modeled as full moment connections...i.e., no pins along the internal lengths of the the ribs since "they" are actually only one continuous piece of steel.
RE: Unbraced Length of Members in a Dome
We are dealing with eight separate roof surfaces of single curvature. If roof deck is considered structurally, the strong axis of each rib at the eight corners of the octagon are continuously braced by deck. The weak axis is braced at about 6' or 7' centers by the octagonal rings.
The major axis of ribs at the middle of the sides of the octagon are not braced for their entire length, either by deck or by the octagonal bracing rings. Those ribs should be removed so that axial load goes to the eight corner ribs and each bracing member should run from corner to corner rib without a central hinge. It serves the dual purpose of bracing the corner rib and supporting the purlins.
BA
RE: Unbraced Length of Members in a Dome
I would like to remove the tubes that are along the face of each curve surface, but I can not. Originally, I only had tubes at the corners, as BA suggests, but, analysis indicated that the lateral movement of the structure, under combinations of dead and wind, as well as dead + live + wind, resulted in lateral movement that I could not accept. I had to add ribs at the middle of each surface in order to get my movement down to a level that was acceptable.
This dome supports loads other that uniform loading. The compression ring also acts as a base for a 18' tall cupola that is to be installed atop the dome (about 9'6" across and 18' tall). So I have to deal with the resulting unbalanced reactions from the cupola. We're only talking a few kips here, but still, its enough to make a difference.
As I have thought this through, my mind comes down to one question......
My ribs work, as they span from the spring point to the compression ring, if the compression ring can be thought of as bracing the curved ribs at the top, thus making the unbraced length of the curved ribs, the distance along the curve, from spring point to compression ring (about 38'). This was my original question......is each rib braced at the compression ring.....thus allowing me to use 38' as the unbraced strong axis length for buckling and determination of allowable compressive stress?
Additionally, I have looked long and hard at google searches, AISC, ASCE, papers etc etc and I do not see where anybody says, "yes, the unbraced length of rib is spring point to compression ring" or "no, its not". I have found several locations where folks have said that there's no data on this type design issue.
There are two issues at work here......what is the unbraced length of the tube rib.....and.....how can we try and work-in linear equations for buckling and allowable stresses to members that are curved, not straight.
I'm thinking the bottom line is.....the ribs need to be reviewed for unbraced lengths along their curvature from spring point to the compression ring until they numerically work, then, the dome needs to have tension rings at three points along the curvature (which really do not help the face of facet ribs), and then the whole thing is clad in metal deck and welded up. This way, it works by the numbers, then has some redundancy, and then has more redundancy.
RE: Unbraced Length of Members in a Dome
I agree with BA.
It made much more sense when I drew a plan view of a typical horizontal octagon with pins at each vertical rib. Then I drew a perpendicular force on one of the joints. You then easily see the instability. You can also see if you could add another horizontal member that would connect between every other rib you could have a truss-type lateral brace system.
Gosh, didn't some engineering professor (maybe it was statics) always say draw a free body diagram of every problem?
RE: Unbraced Length of Members in a Dome
BA
RE: Unbraced Length of Members in a Dome
1. What degree of bracing does the top compression ring provide for the col/ribs or stated differentently
what "K" to use in the rib strong dir. The metal cladding will definitely add regidity to the dome.
I would have to pull the top compression ring with it's different load conditions out of the model
and look at it seperately, preferably with a program that can analyze rings or do it by hand in order
to get a handle on the deflection @ the top of these ribs. This will help in determining what "K" value
to use. I can not claim that this compression ring provides a fully braced point(K=1)for the ribs or
that it is fully translational(k=2) because of the stiffening affect of the cladding. So I would assume
a K=1.5. I would also assume that these ribs are braced in the strong dir @ the horiz trusses.
So the unbraced length that I would assume would be 27x1.5 = 40.5 ft.(this is in the absence of any
accurate theory to the contrary)
I would also make all the ribs the same size.
2. I would have to check and ensure that using the typical AISC column formulas does not result in a
unconsevative result when applied to curvalinear colums.
3. The same applies to the accuracy of the RISA model using segmented lengths to approximate curved
members.
4. Check the stability of the top compression ring and determine the allowable stress to use on it.
RE: Unbraced Length of Members in a Dome
What I see are four rhomboid planes in each of the eight triangular sectors that form the dome above the lower compression ring. All of the vertical ribs that form the arches between the top and bottom compression rings (and there are 16 if them, not 8) seem to me to be laterally braced in the weak direction by the horizontals that form the intermediate tension/compression rings. I think that we all would agree on that. The problem seems to be the concept of being braced or not in strong axis bending.
From my perspective, with or without the triangles of a geodesic dome, there are still inherent vertical components at the transverse member ends that would inhibit strong axis buckling. A concern of whether or not restraint is being provided though, could be made if the horizontal ring members are small in depth when compared to the depth of the main arches, so that the full member depth of the main arch is not engaged. (This is not the case in a geodesic dome where all the main arches, three times the number here, are of the same depth.) Then there could be a valid concern for no buckling restraining in the major axis and the use of the full 27 feet for the unbraced length.
Mike McCann
MMC Engineering
RE: Unbraced Length of Members in a Dome
1) I would tend to use the 6 to 7 ft unbraced lengths that others have suggested. Though I very much understand the confusion that arises in these cases.
2) It is important to model in the curvature between these 6 to 7 ft support points. That way, any compression load will amplify the natural curve in a 2nd order or P-Delta analysis.
3) This is an excellent case where the concept of "unbraced length" breaks down. Shankar Nair has an excellent presentation on the AISC website that talks about why the Direct Analysis Method works better for odd / complex structures where unbraced length is difficult to determine. Essentially the "buckling" failure predicted by the unbraced length methods is really a bending failure produced by the interaction between bending an axial loads. That's why for columns that don't have any nominal bending, then Direct Analysis Method asks you to introduce initial imperfections or notional loads so that the 2nd order analysis will properly amplify these effects and produced the desired buckling.
4) I would strongly recommend the use of the Direct Analysis Method for this type of structure. That way, you can predict the buckling of the structure within the analysis directly.
I believe the link below should take you directly to his seminar... If I have done the link correctly, that is:
www.aisc.org/content.aspx?id=4498
RE: Unbraced Length of Members in a Dome
The radius of gyration of a 16" x 8" x 5/16" HSS is 5.8". If the unbraced length is only 7', then L/rx is approximately 14.5 and buckling is not a consideration and may be ignored.
Does the upper ring provide bracing for major axis bending of each of the sixteen ribs framing into it? The answer to that is no. If the upper ring is deemed to be a rigid body and each rib is hinged to it, the structure is unstable because it consists of eight arches crossing each other, each having four hinges. A four hinged arch is unstable so the structure is unstable.
If each rib is connected to the rigid body with a moment connection then each of the eight arches becomes two hinged. A two hinged arch is stable so the structure is stable.
In Article 7.6 of "Theory of Elastic Stability" by Timoshenko and Gere, a uniformly compressed circular arch of similar geometry to the one under discussion fails at about the same pressure whether it is two or three hinged.
In the absence of intermediate bracing, the unbraced length of each rib might be taken as the arc length of the half arch, or the arc length from spring line to peak of rib extended to midpoint.
For ribs occurring at the corners of the octagon, it could be argued that bracing occurs at 6' or 7' centers. For the shorter ribs at midpoint of the sides, the major axis is unbraced from end to end.
BA
RE: Unbraced Length of Members in a Dome
This is what I have an issue with, and it gets into how the dome is modeled and constructed... "For ribs occurring at the corners of the octagon, it could be argued that bracing occurs at 6' or 7' centers. For the shorter ribs at midpoint of the sides, the major axis is unbraced from end to end."
I can agree if, and only if, there are 8 instead of 16 planes as you go around the dome. At the intersection of the smaller arch and the transverse members, if the transverse members are one straight member, and the arch discontinuous, then I agree with you. However, if the arch is continuous from the top compression ring to the lower tension ring, then there has to be a support component from the transverse member to the arch. It all hinges (no pun intended) on the geometry, as one of the two members MUST be continuous at that joint for the area to be stable.
Mike McCann
MMC Engineering
RE: Unbraced Length of Members in a Dome
As I understand it,
(a) There are 8 (not 16) singly curved surfaces (not planes).
(b) All ribs are continuous from spring line to upper ring.
(c) Transverse members are considered hinged to each rib.
(d) Transverse members are normal to the vertical walls of each of the 8 short ribs.
Because of (d) above, transverse members can brace the weak axis but not the strong axis of the short ribs.
BA
RE: Unbraced Length of Members in a Dome
I guess Marinaman will have to clarify this, because I do not see the same framing you do:
I see:
1. 8 major continuous arches from the top compression ring to the base.
2. 8 minor simple span arch sections between the 8 major arches
3. Three lines of lateral beams that frame from main arch to main arch so which the minor arch sections frame to.
The result is three fold:
1. The major arches are laterally supported in the strong and weak axis,
2. Lateral bracing is not really a major issue for the framing between the major arches as the horizontal members are continuous between the major arches and the minor arch sections just frame to them as simple span beams.
3. The four areas between the major arches are planes, not singly curved.
Really, as I see it, the only truly curved members here are the eight main arch sections.
MARINAMAN, where are you?
Mike McCann
MMC Engineering
RE: Unbraced Length of Members in a Dome
The dome has (8) facets. Each facet's curvature steepens as it approaches the compression ring. The ribs located where facets intersect are curved to a 27'-4" radius. The actual length along the curve of the corner ribs is more like 38'. The height of the dome itself, from the springline to the top is 27'-4".
The horizontals are full penetration welded to the face of the curved tube located down the center of each facet. The horizontals are fastened differently to the face of the rib tubes. I did this so that, in the field, when this is being erected, the horizontal tube can be adjusted slightly to fit, and then be welded into place. The horizontal tube sizes are significant, at 14" x 6" tubes. The attachement of the 14 x 6 tube to the corner ribs is significant as well, but not full pen welded. I'm looking at doing that with 4x4 stiffened angles on each face of tube with all three sides of each angle leg fillet welded to the rib and to the horizontal.
Here are my thoughts at the moment:
- There are two designs here. One for the corner rib.....One for the face rib....as each has a different length.
- The corner rib is braced at the location of each horizontal, in each direction (strong and weak), but, to be conservative, I am taking the full member length of 38' as the strong axis unbraced length and using the horizontals as insurance in my back pocket.
- The face rib is stiffened up in the strong direction by me making the horizontal members big and stiff, and, by making them continuous across the width of facet rib. Even so, I am taking the full curved length as the strong axis unbraced length in calculations and keeping the rigidity added by the horizontals as insurance in my back pocket.
- The "K" value for each rib is somewhere between 1.0 and 2.0. I also believe it to be closer to 1 than to 2. I am considering using K slightly less than 1.5, per my review of how K is derived.
- I'm ignoring the helpful effect of the metal deck.
I appreciate all of the discussion and input from everyone. Its been very interesting and helpful to see and hear the different perspectives.
What do you guys think of these criteria?
Marinaman
RE: Unbraced Length of Members in a Dome
BA...I would assume that the upper compression ring is braced against global translation because of the metal cladding providing trasverse global stiffening of the dome as a whole but could deflect locally in flexture(say ovaling).
RE: Unbraced Length of Members in a Dome
I think your criteria are okay. I don't know what K should be either and I don't think it is a simple calculation to find out but I believe that KL should not be less than the theoretical arc length of the dome from spring line to center point (assuming ribs extended to a theoretical intersection point).
The variable curvature on ribs plus the large amount of field welding is going to make this an expensive project.
SAIL3,
I believe the upper ring is braced against lateral translation by the eight two-hinged arches welded to it. I doubt that the deck plays a significant role in that.
The upper ring is braced against rotational translation about a horizontal axis by the welds between ribs and ring. If these were hinge connections, the upper ring could rotate as a rigid body from the wind force acting on the eighteen foot high cupola. In this regard, it should be noted that full penetration welds cannot develop the full section of the rib.
All of the rings above the spring line (including the upper ring) are braced against rotational translation about a vertical axis by the steel deck. In the absence of diagonal bracing, there is nothing else available to prevent torsional failure of the dome.
BA
RE: Unbraced Length of Members in a Dome
There probably is sufficient bracing there for the 6,7-8' bracing assumption of the main arch ribs. Recall the 5-6% of axial load, rule-of-thumb, being sufficient bracing load for a straight compression member. Your FEA of the total frame should nail these forces down, in more detail. But, that FEA model is really a crappy sketch to see who’s bracing whom, and what the connections and details might actually be. Once you get some of this sorted out, as you are describing in you last post, you probably will find that you don’t really need full pen. welds on every joint, in fact, some of them could likely be field bolted. The thing that strikes me, on this dome structure, is that bracing one arch rib against another is kinda like three drunk sailors leaning against each other for support; and if one moves they will all likely fall over. Be sure this can’t happen.
In terms of stability of the entire dome structure: how much fixed end moment x-x or y-y axis of the ribs can be developed btwn. the ribs and the upper compression ring; what prevents the horiz. plane of the upper compression ring from starting to tip to some plane other than horiz. (i.e. BA’s 4 hinged arch mechanism); what prevents the upper compression ring from rotating torsionally about the vert. center axis of the bldg. and causing a torsional/rotational crumpling of the entire dome structure? The geodesic dome system controls many of these movements with individual members (struts) primarily axially loaded and the triangular shapes of each small plane surface of the structure prevents the racking or parallelograming of each of these shear planes (tension fields/surfaces).
You might want some x-bracing (k-bracing, as in your lower walls) btwn. some of the node points on the adjacent ribs to provide this torsional stiffness, and/or to force the development of some end fixity in the ribs, w.r.t. their weak axis, almost as a knee brace would induce this bending loading or stiffness. This bracing can be provided within the same depth as the main ribs. Unsymmetrical gravity loads and lateral loads (wind, EQ, cupola, etc.) certainly will exacerbate these problems. I also think the proportions of the upper compression ring are bass-acwards, it should be about the depth of the main arch ribs, and much wider, rather than thin and deep as shown on your model.
RE: Unbraced Length of Members in a Dome
Interesting thread. I like how engineers are smart.
RE: Unbraced Length of Members in a Dome
I read through most of the commentary and gleaned what I could, especially BA and Dh's last comments. I decided to take apart the dome and and flatten the pieces out into 2 D frames (eight of them), it is a little simpler for me to get my head around.
So your Lu would become the intermediate horizontal member spacing, with each adjacent frame/panel providing the bracing against the buckling force of each rib. In my sketch I was mainly looking at out-of-plane (of the dome) buckling of each rib, or strong axis of the HSS ribs.
You can also visualize how lateral bracing due to wind or seismic would work, which will obviously end up being a much larger force to design for then the buckling force.
I reserve the right to be totally off on all of this, but I feel that is what most of you are saying?