Moment redistribution - Concrete Continuous Beams

[7788_011]

Given that for continuous concrete beams the moment generated at the supports will normally exceed the cracking moment Mcr of the beam, isn't it a must to increase the design bending moment & corresponding reinforcement at mid spans generated from a elastic frame analysis?
Can anyone please tell me how you consider this in you design? Do you compare the negative moment with Mcr and simply apply the positive moment at mid spans by a amplification factor （what factor is adequate if so?) or do you manually reduce the stiffness of the beam segement at supports in your analysis?

 

[centondollar]

Elastic structural analysis and plastic sectional design of rebar is always a lower-bound solution, i.e., a "safe" option regarding design for strength.

Remember that hairline cracks will occur immediately upon significant loading along the entire beam (assuming that it is designed correctly, i.e. is under-reinforced and not thick enough to remain uncracked under normal loading), which means that stiffness will drop along the entire beam. This implies that deflections increase, but the moments do not, since there is no "stiffer part" for the load to migrate to after each section has proceeded into the cracked stage.

Furthermore, if we assume elastic analysis to be reasonably correct for a beam that has hairline cracks, the total strain energy is constant and therefore, the moment cannot increase without increase of external load. Think of the clamped-clamped and simply supported beams: both have the same area under the moment diagram, but the diagram is shifted and the magnitude of the shift depends on boundary conditions.

 

[7788_011]

Okay thanks.
If I am not mistaken, as long as the beam doesn't become statically determinate (ie not hinge generated at supports) we don't need to consider moment redistribution for positive moment because once it exceeds the MCr+ it will redistribute back to elsewhere?

 

[rb1957]

"statically determinate" ? your sentence looks backwards to me ...

statically determinate would mean creating hinges at each support (if hinges means simply supported joints.

your proposition seems "nonsensible" to me ... if you have a continuous beam, then moment distribution is the correct approach. If this analysis generates large moments at the supports, exceeding Mcr, then I'm assuming this means that the beam cracks (not yields) ... which I'd've thought was a bad thing. If you can live with this situation, then you have to return to your moment distribution, and change the "failed" support to a hinge (simple support) and recalculate.

another day in paradise, or is paradise one day closer ?

 

[HTURKAK]

Say ,The reverse is true.. when negative moment exceeds the MCr- , then it will redistribute to the span moment ..

Assume that, a continuous beam with several spans and assume that , fixed end moment can develop at one of the span ( could be acc. to span dim's and loading )
When you idealize that span with a single span with fixed supports ;

- If you do elastic analysis , Ms-= q*l^2/12 and Msp+=q*l^2/24 you may design and select the flexural reinf . acc to these moments and total moment will be Mt=q*l^2/8,
- If you do plastic analysis Ms-= q*l^2/16 and Msp+=q*l^2/16 and still total moment will be Mt=q*l^2/8,
- If you assume simply supported beam Ms+= Mt=q*l^2/8..

One can notice that the total theoretical flexural reinf. is the same for three conditions.

When you do incremental load test a continuous beam ( designed for elastic moments ), the initial cracks will develop at supports then moment redistribution will start and eventually the final moments would be elastic moments..

The reason for elastic analysis and designing for the elastic moments is, essentially to obtain minimum deflection.

I hope this respond answers your question..

 

[rb1957]

does the concrete tend to fail in tension, and then the beam rebar continues to carry the tension ?

another day in paradise, or is paradise one day closer ?

 

[Lomarandil]

RB1957, as the tension side of a flexural member is loaded, the concrete and steel are both strained (as they are bonded together).

For the concrete, this strain results in a series of small cracks crossing the section. In a properly detailed beam, they are well distributed and small (less than 0.4mm).

At the same time, the strains are developing stress, and ideally the yield stress, in the steel closest to the tension face. Because rebar has a nice long stress-strain curve, the beam can happily carry load with the bars yielded as the tension force in the moment couple.

----
just call me Lo.

 

[HTURKAK]

The tension strength of concrete 5- 10 % of compression strength and one of the basic assumptions of the RC design is, neglect the tension strength of concrete .

 

[7788_011]

Understand the total of WL2/8 as a whole will not change. I guess what I tried to say what I initiated this was, say use your example, I have different reinforcement for span and ends and if I design to the elastic case with ql^2/24 for my design positive moment, and when it cracks over the ends, the positive bending moment will increase to upto WL^2/8 (when they become hinge at ends though it wont happen as we design the end to WL^2/12 in the first place) then it might be under designed for the span. But I just realized that when the redistributed positive moment increase to Mcr value it will be redistributed to elsewhere again and when designing we design to plastic section as centondollar pointed out so the capacity it will be way greater than Mcr... Overthinking here

 

[Celt83]

Info for the following plots:
Spans- 10ft (4 total so model is symmetric)
Column heights - 10 ft
Column far end fixity - Fixed
Columns assumed not to be compressible

f'c = 4 ksi
Beams - 12x24 with (3)#8 top and bottom
Cols - 12x12 with 1% As

Beams loaded with a UDL of 1 kip/ft

Gross Moment of Inertia for all members:

Beams with 0.35 Ig:

Exterior Beams with 0.35 Ig everything else Ig:

Interior Beams with 0.35 Ig everything else Ig:

Beams 0.35 Ig and Cols 0.7 Ig:

I'm making a thing:

http://www.thestructuraltoolbox.com

(It's no Kootware and it will probably break but it's alive!)

 

[KootK]

But I just realized that when the redistributed positive moment increase to Mcr value it will be redistributed to elsewhere again and when designing we design to plastic section as centondollar pointed out so the capacity it will be way greater than Mcr... Overthinking here

It sounds to me like you've got a pretty good handle on this and, for the most, did from the start. In general, things proceed like this:

1) For low load levels, nothing has cracked and your elastic stress distribution is fairly valid.

2) Often, your hogging, support moments create the first flexural cracks. Then, until some sagging, mid-span moments develop, the middle of your beam is disproportionately stiffer than the ends relative to your initial elastic assumptions. So, during this part of the load history, the mid-span moments are getting taxed more that an elastic analysis would predict, just as you suggested originally.

3) Eventually, the sagging, mid-span moments also create cracks and things substantially revert back to your original, elastic analysis informed, plastic design assumptions. This is just as you said in your last comment.

All that said, this is far from perfect. Consider:

4) Your supports are likely to always be more cracked than the mid-span.

5) large segments of your beam between the support and mid-span will likely remain uncracked and these affect load distribution as well.

6) Many modern codes contain provisions limiting the amount of redistribution available based on how heavily reinforced the beam is. There are limits to how much redistribution is possible before high, local strains fracture the bars. They don't just stretch happily forever. The more heavily reinforced your cross section is, the less redistribution is generally possible. And, as you know, beams tend to be quite heavily reinforced at continuous supports.

7) It's unrelated to cracking but there is a dimension in which sagging moments are safer than hogging moments. Hogging moments are usually dependent on what is going on in the adjacent span. Are the loads assumed to be there, really there? Have accurate assumptions been made about the cracking in those adjacent spans? Who knows...

Can anyone please tell me how you consider this in you design? Do you compare the negative moment with Mcr and simply apply the positive moment at mid spans by a amplification factor （what factor is adequate if so?) or do you manually reduce the stiffness of the beam segement at supports in your analysis?

1) We don't do much, frankly, other than to accept that structural engineering is really just the moderately intelligent, course proportioning of things. There's not much "knowing" involved.

2) Where redistribution limits apply, we check them for the final ULS state.

3) I've known several, brilliant, older engineers who have recommended amplifying the sagging, mid-span moments just as you've suggested. I never design my mid-span for anything less than WL^2/20 as a result. This is not very scientific.

 

[centondollar]

788_011,

You misunderstand me. For any structure, statically determinate or not, the initial elastic internal forces (bending moment, shear force, normal force) will equal the internal forces after the beam has cracked in every cross-section.

During the cracking process (load attracted from cracked sections towards uncracked sections, "force follows stiffness"), maximum moment will never exceed the elastic moment, because as soon as moments increase locally (due to the cracking of adjacent sections) and exceed cracking moment, the section will crack, causing rebars to be in tension and concrete in compression (plastic design of the section) and for the cracking to migrate to the next section that is not yet cracked. This process continues until either:
a) everything is cracked, which leads to a larger deflection but internal forces equal to the elastic solution (relative stiffness of members is once again identical)
b) many cross-sections are cracked, and the ones that are not cracked have sustained a moment that is below the cracking moment and thus no issue arises

Applying stiffness modifiers (which, when done according to standard, is more or less arbitrary and does not reflect the actual nonlinear response of a concrete frame) to complicated frames and then making multiple calculations will show you a simplified version of the abovementioned story.

 

[7788_011]

Thanks everyone.
@centondollar What I tried to say by "statically determinate" is that say a simply support beam, there is no moment distribution involved.

A bit off topic but I am wondering do you reduce your out-of-plane stiffness of RC walls that connect to RC slabs to say 0.1 or even less in your analytical model for gravity loads to avoid extensive reinforcement?

 

[7788_011]

I am still a bit confused.

I have this from CSI website where it says "Walls are generally not designed for out-of-plane bending to avoid excessive longitudinal reinforcement. In this case, use a small modifier say 0.1 for m11, m22 and m12 so numerical instabilities could be avoided. "

https://wiki.csiamerica.com/display...cked+section+properties+for+building+analysis

Now lateral aside, for gravity, if we reduce the out-of-plane stiffness of the wall, letting the moment redistributed to slab spans, then do we crack the slab the same way? If so, then pretty much like what we discussed above, the out-of-plane moment @ walls will be the same as uncracked condition, then we have the same moment to design to so we are not able to "avoid excessive longitudinal reinforcement" to begin with. Thoughts?

 

[KootK]

..then do we crack the slab the same way?

Yes except, in all likelihood, you probably do exceed code limits on redistribution.

...then we have the same moment to design to so we are not able to "avoid excessive longitudinal reinforcement" to begin with.

I don't believe that their is a theoretical justification for this other than we deem it to be of relatively little importance. Again, it comes down to this:

We don't do much, frankly, other than to accept that structural engineering is really just the moderately intelligent, course proportioning of things. There's not much "knowing" involved.