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The lower plastic limit theorem is stated in Wikipedia as follows:

If an equilibrium distribution of stress can be found which balances the applied load and nowhere violates the yield criterion, the body (or bodies) will not fail, or will be just at the point of failure.

I'm particularly interested in this theorem in the context of reinforced concrete.

On this website, design of concrete corbel is described using "Strut and tie" method, which applies the lower plastic limit theorem.

In this method, the concrete joint is assumed to consist of struts in compression, representing parts of concrete in compression, and ties, representing reinforcing bars in tension. The struts and ties are then modeled as a truss, with members in compression and tension.

enter image description here

So, at the heart of my question is this: Do we require previous knowledge about the load paths through the structure before we can apply the lower plastic limit theorem?

The theorem as stated on Wikipedia states that if an equilibrium distribution of stress can be found which balances the external loads which will not fail, the structure will not fail.

So if we first just come up with a placement of reinforcing bars, with concrete assumed to be balancing this "truss", and the rebar does not yield, nor the concrete compressive stress is not exceeded, will our structure hold (not taking into account that it can be unoptimal)?

Considering the picture above: we have placed a bar close to the top and assumed that the locations where we have the compression struts (concrete in compression) are locations where the concrete will and can actually be in compression. Would any such arrangement be correct, assuming the "truss" will not fail? Or do we need some knowledge beforehand that such arrangement is actually close to the real distribution of stresses in the structure?

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  • $\begingroup$ Check out the work done by Barnes Wallis on how concrete behaves. $\endgroup$
    – Solar Mike
    Apr 15 at 11:40
  • $\begingroup$ The wiki didn't about its applications but says the lower plastic limit theorem applies to any elastic-perfectly plastic body or assemblage of bodies. Neither is close to the reinforced concrete. $\endgroup$
    – r13
    Apr 15 at 13:32

2 Answers 2

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To answer your question: yes, absolutely any reinforcement scheme which satisfies the criterion will withstand the loading without failing.

Emphasis on "without failing". A poorly thought-out reinforcement layout may resist the applied load, but it will do so poorly. It won't fail according to ULS (ultimate limit state), but it won't get anywhere near satisfying the SLS (serviceability limit state).

So it won't collapse, but it will have cracks a mile wide.

Thankfully, coming up with reasonable tie-and-strut layouts is pretty straightforward, so the reinforcement you adopt will likely be somewhat near optimum.

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  • $\begingroup$ Quite interesting. I have now two answers that seem to contradict a bit. Kamran suggests that wrong placement of ties could impair the structure's stiffness.. I wonder why serviceability is not necessarily satisfied. Shouldn't the stiffness of the equivalent truss be similar to that of the real structure? I mean, if the strut and tie truss is designed to not displace in a way so that it does not violate SLS, wouldn't the real structure also satisfy the same requirements? $\endgroup$
    – S. Rotos
    Apr 20 at 10:57
  • $\begingroup$ @S.Rotos: The issue is that a lousy layout will imply a lousy strut-and-tie. For example, imagine you move the rebar down to the middle of the corbel (effectively moving the TTC and CCT nodes down). This would be a terrible choice, obviously, since the diagonal strut would be under much greater compression and the tie, under greater tension. But let's assume the concrete and rebar is strong enough to withstand it anyway. So ULS is satisfied, the structure won't fail... But what about all the concrete above the tie? It'll be under significant tension, which actually means it'll crack all over. $\endgroup$
    – Wasabi
    Apr 20 at 19:39
  • $\begingroup$ @S.Rotos: Strut-and-tie makes no reference to SLS. It is only meant for ULS. I believe SLS is assumed to be satisfied since a reasonable strut-and-tie is conservative, so you're placing more steel than is necessary and therefore probably satisfying SLS as well. And as for kamran's comment on stiffness, he's correct. In my example, moving the rebar down to the middle of the corbel will make the corbel's effective height (from rebar to opposite fiber) much lower, reducing its moment of inertia and therefore stiffness. It can withstand the load, but it'll be far more flexible while doing so. $\endgroup$
    – Wasabi
    Apr 20 at 19:50
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Ultimately the answer boils down to wether the geometry of your structure combined with particular loading will confirm to the imposed behavior by preplacing the rebars assuming the truss stress pathway or not.

There are cases where placing extra reinforcement at an incorrect location can lead to imparing stifness and stress concentration, and even failure in long term, especially in cyclical loading and unloading situations.

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  • $\begingroup$ Interesting, I wonder how does extra reinforcement impair the stiffness? $\endgroup$
    – S. Rotos
    Apr 20 at 10:54
  • $\begingroup$ @S. Rotos, imagine a cantilever beam and then add extra reinforcement at the free end. you deny it the flexibility to bend. it will transfer a concealed stress to the support. $\endgroup$
    – kamran
    Apr 20 at 11:13

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