post tension zone of influence

[structSU10]

I am investigating a PT garage that has some failed tendons throughout. I have been getting a handle on the amount of PT loss that is still acceptable to performance of the PT slabs on the basis of minimum effective post tensioning force to meet service and strength requirements. The one thing I am not sure on is how to determine the effective width where losing tendons becomes an issue locally.

For example if the floor had 15 kip/ft effective post tensioning as designed and can safely perform with only 10 kip/ft, and tendons are spaced at about 2' on center and I have lost 3 tendons near each other but the remainder across a width are enough to provide the 10 kip/ft required, what is a reasonable criteria for a zone of a few tendons close to each other that may be broken? My thought is Poisson effect makes the overall compression spread pretty well across the width of the slab past a couple feet from the edge so perhaps the compression stresses can be assumed uniform but the reinforcement provided by PT is investigated locally? I could check a zone thats 4-6 times the slab span for the reinforcing present to check for adequacy. Does this sound like a reasonable approach or are there better methods?

 

[Ingenuity]

Sorry I thought I broke the E-T internet with this error last night:

So for the following two cases they have the same ultimate moment capacity?

Usual qualifiers like:

(i) Span is sufficiently long enough that it is flexural behavior (no tied-arch for CASE I, no arching for CASE II)
(ii) Axial pre-compression is the same magnitude in both cases
(iii) Neglect area of PT and duct/sheath in CASE I
(iv) No buckling for CASE II
(v) No secondary effects for CASE II​

 

[KootK]

So for the following two cases they have the same ultimate moment capacity?

I would say yes, assuming that you've got some dead & live end anchorages in case 1 rather than the tendon just passing through to parts unknown.

 

[Ingenuity]

...assuming that you've got some dead & live end anchorages in case 1 rather than the tendon just passing through to parts unknown.

Yes, graphically there should have been an anchorage at each end.

So similarly, for an intermediate CASE 1a (2 each no drape unbonded tendons, same total pre-compression at other 2 cases - see below) it too would have the same ultimate moment capacity?

 

[centondollar]

Obviously not, since the tendon is eccentric from the centroid of the cross-section.

Stress = (M*y)/I + P/A = (M*(distance_of_tendon_to_centroid))/I + P/A (with appropriate and consistent sign rules)
In case 1a (figure in the previous post), y is nonzero, and thus the capacity affected by the sum of bending and compression action.

EDIT: Wrote this in a hurry, and noticed that ULS - not SLS - was the subject matter. Regarding SLS, the case 1a has only precompression, assuming that both tendons are at equal distance from N.A. (M/W term vanishes, due to opposite effects of the straight, eccentric tendons).

 

[KootK]

So similarly, for an intermediate CASE 1a (2 each no drape unbonded tendons, same total pre-compression at other 2 cases - see below) it too would have the same ultimate moment capacity?

I vote yes for case 1a having the same ultimate moment as the other cases. This is fun. I feel as though I'm in the witness box being slowly led into a trap by a crafty back country lawyer. A Mathew McConaughey character maybe. I look forward to soon being hoisted by own petard!

 

[rapt]

Koot,

I have stayed out of this to avoid reigniting your fears about the Australian engineering mafia. There is no set up from what I can see. Ingenuity's questions/comments are all referring to your comment below. He is trying to get you to rethink your (4) based on these examples.

4) In unbonded PT, the compression -- and not the strand itself -- IS the reinforcement.

All 3 of his cases have the same area of prestress strand and the same P/A.

1 and 1a do not provide any uplift.

2 has uplift from a moment couple due to the eccentric anchorages, not from the curvature of a draped tendon.

So if there is full restraint in all 3 cases, there is no P/A and no Mp. If there is no restraint, 1 and 1a have purely P/A and no Mp.

If you add case 3 with the same number of tendons draped as a parabola or a harped tendon from the centroid at the anchorages to a low point at midspan, then at midspan it will have P/A and Mp equal to those in 2. But it also has uplift from the curvature of the tendons, so if fully restrained will still provide Mp as a prestress moment.

They all have different ultimate moment capacities according to the way Ultimate Capacity is calculated for unbonded prestress, except for 2 and 3 at the point in 3 where the tendon is at maximum eccentricity. They all have the same compression P/A. Only one actually has upward curvature induced by tendon curvature.

 

[Settingsun]

Rapt, what are you referring to as Case 2? Ingenuity's Case II doesn't have eccentric anchorages, or any anchorages for that matter. It's externally compressed on the centroid.

 

[rapt]

Sorry, I was mixing cases without looking back at the previous posts.

My case 2 is a straight tendon at minimum bottom cover.

Ingenuitys case 2 has no ultimate capacity at all.

 

[KootK]

I have stayed out of this to avoid reigniting your fears about the Australian engineering mafia.

1) It's not a fear; it's simply an observation. And a demonstrably accurate one at that. I enjoy these conversations immensely, especially when the Australian PT Mafia shows up.

2) I welcome your contribution on all PT threads. In fact, I was wondering why you hadn't surfaced here and was considering asking Ingenuity to summon you. Please never, ever hold back from contributing on a thread for my sake. I don't need that, I don't want that, and our community would be poorer for that.

There is no set up from what I can see... He is trying to get you to rethink your (4) based on these examples.

Agreed. The humor that I had tried to interject with the back country lawyer bit had nothing at all to do with my fearing being ambushed by a bunch of Australians. Rather, I could simply see that Ingenuity was carefully crafting a chain of logic that would eventually lead me to, hopefully, some intellectually fertile new ground. I attempt that myself with some regularity and appreciate it when others do the same for me.

My case 2 is a straight tendon at minimum bottom cover.

I'm still confused. Your case2 two would seem to match none of Ingenuity's cases. Did you mean for that to be a separate case for us to consider?

Ingenuitys case 2 has no ultimate capacity at all.

I disagree. I believe that case 2 two does have an ultimate capacity and that the primary things that it lacks are:

3) An ultimate capacity based on the yielding of composite reinforcement. But, then, that is true of all unbonded PT without the addition of bonded reinforcement.

4) The ability to deviate the tendons to match the displaced shape of the beam as shown in the sketch below. And that's really what I was getting at with this earlier statement:

In a ULS event taken to the extreme, where caternary-ish stuff starts to develop, yeah, there's a difference. However, to my knowledge, we don't normally consider that in ULS flexural design calculations.

 

[KootK]

If you add case 3 with the same number of tendons draped as a parabola or a harped tendon from the centroid at the anchorages to a low point at midspan, then at midspan it will have P/A and Mp equal to those in 2. But it also has uplift from the curvature of the tendons, so if fully restrained will still provide Mp as a prestress moment.

I feel that the italicized portion of that statement is incorrect in a fundamental way. As shown in the sketch below, I believe that an unbonded, post-tensioned beam with its anchorages at mid-depth derives its flexural capacity from two actions acting in concert:

1) The concentric pre-compression induced at the ends by the anchorages and;

2) The balancing force induced by the curvature of the tendons.

A member needs both of those things, acting in concert, to develop its full flexural capacity. If you remove the concentric pre-compression but keep the balancing forces, then the member will have some flexural capacity but, in most cases, not anything close to its full flexural capacity. That's what I was getting at with this statement:

1) Not quite as you've still got the load balancing effect of the tendon drape. The end product sort of resembles a suspension bridge rather than a prestressed concrete member though.

.

The balancing load contribution can be envisioned in at least two different ways with respect to its contribution to flexural capacity:

3) As a transverse, uplift force combined with a pre-compression force concentric with the anchorages (green arrow below) or;

4) As shifting the pre-compression force from the elevation of the anchorages to a location concentric with the pre-stressing tendons (blue arrow below).

Obviously, #4 is what we do in practice. The salient point here being that it's one approach or the other. Combining the balancing load with the effect of a deviated, unbonded tendon represents double dipping. Conversely, there's no way remove the pre-compression without sacrificing capacity.

Dialing things back to a more fundamental, Newtonian physics perspective where conservation generally rules: does anyone really think that we could remove the pre-compression effect from a prestressed member and not have that cost us something? I view that as being self evident with or without tendon drape.