Strut-and-tie modelà once again,
Strut-and-tie modelà once again,
(OP)
Please bear with me on this. It was nine years ago when I had my first exposure to the STM, yet never use it until recently.
Referring to attached sketches, the compression force in the strut BC is a result of the tension in the tie AB. Now, we know that stresses in the tie decrease with the development length of the bar to zero after an ld length, yet the STM assumes constant bar stresses. Doesn't that lead to wrong results? Is not the force in BC in sketch 2 zero, while the STM gives us a force= T(AB)/ cos theta2?
It appears that I am missing something here.






RE: Strut-and-tie modelà once again,
My understanding is that member BC needs to take compressive force if member CD is a tie in your model. Through statics, if you did not have member BC taking compression you would blow out the back of your column at D since CD would become a compression member and probably lacks sufficient lateral restraint.
Hope this helps.
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
Let me try to explain, but don't give high hope though.
The strength of bar AB is developed by bond, and the bond is usually assumed uniform throughout the development length "Ld". In turn, Ld is determined by expermiments on the instance that yielding of steel bar is imminent, and the bond starts to fail. The imminent failure causes the bar to slip (uniformly), thus the bar feels the same stress everywhere along it's length, provides as Hokie pointed out, the anchorage and end nodes are adequate to alow the stress to develop.
Your thinking is not necessary in the wrong path, just the present time research has not tilted your way yet. One day it could be though. (ie. I had similar question before, but gave up after struggled a while along the road. I don't think we are loners on this matter)
Suggest to take the face value of the code for now.
RE: Strut-and-tie modelà once again,
I'm presently prototyping some concrete D-regions with the strut and tie model but cannot figure out how to anchor the ties meeting at the nodes. Typically, how are ties anchored at the nodes in strut and tie members? Are headed bars my only alternative? Where can I find a few real world examples of such details?
Thanks.
RE: Strut-and-tie modelà once again,
There are a number of ways to anchor the tension bars in strut and tie models. Headed studs are sometimes used, bars are welded to embedded plates, and hooked bars work in certain applications. In Ailmar's example above, the AB bars should be detailed as horizontal ties and developed by continuity around the column verticals, not by bond along the length.
RE: Strut-and-tie modelà once again,
BA
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
The A-B-D bars are not restricted to just two bars. There could be four or six bars. The anchor bar running horizontally just inside the ninety degree bend is the best method of anchorage and puts the compression right into the middle of the strut.
BA
RE: Strut-and-tie modelà once again,
If tension must be considered constant because of slipping, then the whole idea of developement length is not valid.
RE: Strut-and-tie modelà once again,
You have a strange way of viewing this situation. It is unrealistic and unsafe to assume that bars in strut and tie models develop their capacity by bond. In general, they do not! So, if they do not, it is necessary to develop the bar force in some other way, namely by adequate anchorage. It is conservative to ignore the small contribution of bond in favor of a more positive anchorage.
The "whole idea of development length" is valid if there is enough length to develop the bar. Otherwise, the bar must be developed in a different way. Your last post suggests that you do not understand or are not prepared to accept the concept of the strut and tie model.
BA
RE: Strut-and-tie modelà once again,
I didn't read Ailmar's last post that way. It seems to me he understands that anchorage is essential rather than bond, but he has just not expressed it well. But then maybe he doesn't understand, because he says kslee made it clear to him, and kslee's post only confused me.
As to our different way of detailing a corbel on a column, just as the corbel bars are not restricted to just two, neither are the column verticals.
RE: Strut-and-tie modelà once again,
I am confused as you are as well :) :) :)
Let me re-state my original response with an example:
For a #4 bar, As = 0.2 si, Ds = 0.5 in, Ld = 15 in (30*Ds), Tu = 12k (at yield).
My claim is simply B (bond) = Tu/Ld = 12/15 = 0.8 ksi (800 psi) uniformly throughout the bar, as contrary to the notion that the bond stress is max at the start of the development length (2*0.8=1.6 ksi), then diminishes to zero at the end of bar (B = [1.6+0]*15/2 = 12, check too).
The uniform distribution of bond stress in under the premise that at the imminent of yielding, the slip mechanism in between bar surface and concrete is engaged to develop the yield strength, as domino, both the bar and the concrete feel the smae stress uniformly throughout the development length.
For a hooked bar, the mechanism is much more complicate. No comments on that.
Ok, guys, get out your guns & bullets ready, aim, and shoot :)
RE: Strut-and-tie modelà once again,
On case 2, say the length of bar AB >> Ld, is there a reaction at node B? If no reaction at B, is bar BC required? If not required, what is, other than CD, required to take on the force component from AC?
Please comment & explain.
RE: Strut-and-tie modelà once again,
If you do not wish to rely on the anchorage of ABD into the column, you have the option of using strut BC. If you do that, you are required by strut and tie procedures, to develop the bar at node B assuming no bond between A and B.
BA
RE: Strut-and-tie modelà once again,
The question here is if AB alone is MORE THAN adequate, why do ABD?
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
If bar AB is fully anchored in the column with stress tapering down to zero at node B, there is no strut BC with a known dimension and a known compression. The state of stress in the concrete around the bar is not precisely known but there is a compression field in the vicinity of the bar. The tension in bar AB, together with the horizontal component of strut AC puts moment into the column of magnitude Vu*e where e is the eccentricity of load from centerline of column.
Looking at sketch #2, I think I would be more concerned about developing the 'A' end of AB than the 'B' end.
BA
RE: Strut-and-tie modelà once again,
Please check below.
By inspection, you have Tension in AB (from max at A -> 0 at B), constant Compression in AC & CB. Then the compression in CB at B triggers tension in BA & BD. So, now node B becomes an anchorge point to develop the required tension for balance. Does this make sense?
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
Why the tension in AB is the same along its length and why bond doesn't come into it? Tie AB is not installed in a greased sleeve.
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
It should be clear to all that this is a failure waiting to happen even if bar AB is anchored with three times the required development length. The failure surface will be below bar AB and above the column verticals.
Bar AB must be anchored at A and B. It requires a hook and a vertical leg below node B to transfer tension to the column reinforcement. The location of compression struts is not an exact science. When using the strut and tie model, the designer is free to modify the location and slope within certain limits.
BA
RE: Strut-and-tie modelà once again,
I agree with your latest response, that the existence of bond does not have explicit role on design theory for this case. But the bond is there does the work - develop tension in bars, only we/code opted to ignore it for the simiplicity of design process.
Ailmar:
Maybe you should try to get papers on the development of trut-tie method to see the full context. For the time being, as pointed out by Hokie, there is no harm in ignoring the bond-steel interaction in this (corbel tension bar design) regard.
RE: Strut-and-tie modelà once again,
Regarding the original question - in a case such as this we have a "d" region (or disturbed region) where plane sections are not plane after the load is applied = non-linear strain distribution, rather than a "b" region (or beam region), where strains are linear thru the depth and plane sections remain plane.
For example, in a deep beam, with a point load applied at the top (in the middle of the beam), there must be a tension tie at the bottom of the beam (like a tied arch). If you cut a section thru the beam near the support and near the middle, the bar tensile strains would be almost the same value. In other words, there is roughly a constant amount of tension on almost all of the length of the bar (the tension develops very rapidly from the thrust of the diagonal compression struts), b/c of the non-linear distribution of strains.
This is why it is critical to develop the bars very quickly, sometimes quicker than the development length allows for, so in this case you need use mech. anchorage at the end of the bar sticking into the corbel.
So for your corbel situation, you should weld a cross bar at the outside face of the bars running into the column (to provide immediate anchorage of the bars beyond the column face). Your other option is to weld the corbel tension tie bars to an angle, on the outside edge of the corbel.
In addition, you should provide the minimum ldh (or embedment length req'd to develop a hook in tension) running into the column. This is measured from the face of the column to the back edge of the tails (near the back edge of the column). If you have less than ldh, prorate your area of tension-tie steel.
Be very careful when using ST models - if you go down the wrong road you can have an unsafe design - there is very little re-distribution of stresses after cracking. If i were you, I would use the empirical method in the PCI Manual - you will always end up with a safe design and safe details + corbel proportions that way. (Attached is a spreadsheet I wrote for the Cdn code, but you can modify for the ACI code as well - use at your own risk).
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
Very impressive piece of work. I will need to spend a little time digesting it.
BA
RE: Strut-and-tie modelà once again,
RE: Strut-and-tie modelà once again,
Please bear in mind, the design of reinforced concrete is not an exact science, rather, it is based upon ongoing researches and test results. All design methods are considered the "best match" with logical assumptions to allow for ease of understanding and use.
You started a good question, wish you have gained something after this discussion. If there is anything unclear, keep asking, or do more reading on materials that are relavent to your question. But do not confuse design with research.