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Large Deflections in Concrete Slab Without Cracking
6

Large Deflections in Concrete Slab Without Cracking

Large Deflections in Concrete Slab Without Cracking

(OP)
I have an unusual issue on a project where there is a concrete slab that is deflecting 2.5" in the worst location only under construction loading. The two-way slab was designed using RAM concept and from the results, there should only be 0.6" of deflection occurring under construction. The area in question is a 33' span, the slab is 10" thick with an f'c of 5,000psi.

I've gone to the site and confirmed the deflections but strange part is there was no visible cracking on the top or bottom of the slab. The slab was poured all in 1 day and there were no visible construction joints in the slab either. I also noticed that there were larger than expected deflections throughout the slab, not just at the worst case condition. At 1st we thought that the contractor may have pulled the forms off to early as they pulled them 15 days and didn't reshore but the concrete break tests indicate roughly 4,000 psi at just 7 days (High-range ware reducers were used in the mix). It's also worth noting, the slab was poured 6 months ago and then there was a pause on the job over the summer so the 2nd story slab has yet to be constructed. The contractor stated that there were no materials that were stored on the slab over the pause and they noticed the deflections when water would begin to pool when it rained. Based on the inspection reports everything seemed to be correct regarding the rebar and pour. We have asked the contractor to reshore the entire slab until the source of the problem can be found.

Due to the lack of cracking my current theory is that there may have been an issue with the formwork while the slab was being poured that resulted in the slab deflecting before it ever started to cure. The issues is this doesn't fully explain why the deflection can be seen on the top and bottom of the slab or why the larger deflections are consistent throughout. If we can't determine the source our next step is to try and have testing done on the slab to see if there is a discrepancy between the slab and the concrete core results. Does anyone have any suggestions on what else could be the cause?

RE: Large Deflections in Concrete Slab Without Cracking

Over 10 meter span (in two directions, no less) with only 250mm uniform slab thickness is quite a feat, and it's no wonder that it's deflecting on the order of L/200 (10 000 mm / 50mm = 33 feet / 2,5 inches) with those dimensions. Whoever predicted roughly 12mm deflection based on software results must have fudged the numbers: forgotten to reduce elastic moduli due to creep, forgotten to properly account for prestressing losses, or forgotten something else, such as realistic boundary conditions.

What do you define to be "construction loading"? The loads add up quite fast if you do the math: one square meter covered with cement bags instantly adds approximately 3,5 kN/m^2 (25cm height, 14 kN/m^3 density for cement) of uniform live load, with tools, topping concrete and props in particular adding quite a lot more load to the slab.

Regarding the minimal cracking, I suggest that you calculate the deflection with a simple beam formula for a 1m wide strip and use elastic moduli reduced due to creep and 2nd moment of area with uncracked transformed section. This will give you the order of magnitude of deflection expected at the given loads for a prestressed and mostly uncracked slab. If the results are far off from 0,6 inches, the error is probably in design.

RE: Large Deflections in Concrete Slab Without Cracking

Did you check for cracking. Higher strength concretes tend to behave more elastically than plastically

So strange to see the singularity approaching while the entire planet is rapidly turning into a hellscape. -John Coates

-Dik

RE: Large Deflections in Concrete Slab Without Cracking

Is this a PT slab or conventional reinf? If it is a conventional reinforced slab your span/depth ratio (40) is quite large. Is the deflection you mention an interior span or other?

RE: Large Deflections in Concrete Slab Without Cracking

No matter how you look at it, or whatever your computer model says, your slab is too thin. L/D = 39.6 should have been a red flag.

RE: Large Deflections in Concrete Slab Without Cracking

But the computer said it would be fine! pc3

Quote:

The two-way slab was designed using RAM concept and from the results, there should only be 0.6" of deflection occurring under construction.

RE: Large Deflections in Concrete Slab Without Cracking

Tough crowd.

You mentioned this is the first of two levels so assume this is not a transfer slab and maybe a parking or residential level. 10” is absolutely on the high end for L/d but not outside the realm of working for a lightly loaded residential slab, assuming we are talking flat slab with drops not a flat plate.

Some things to consider:
Cracking can occur at the micro scale and reduce stiffness, so just because you didn’t see surface cracks doesn’t mean the concrete isn’t cracked. Any separation at all removes the ability to pass tensile stresses through the concrete.
You mentioned a 2nd story slab does you model have the columns up included, that would significantly increase the slab column joint stiffness
You mentioned the slab was constructed 6 months ago and has sat idle. I assume this means unprotected from the elements, which means your slab as likely seen 100 degree ambient temp swing (meaning much higher on the sun exposed surface). (You can put a thermal load in concept but will need to do manual combinations to capture that effect plus the self weight)
At 6 months you should be considering some creep deflection as well.

Factoring in all of that plus some likely formwork deflection and differential settlement of supports and 2.5” sounds in the ball park.

I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

These are the sort of results I have encountered previously when comparing calculations calculated using software that does not account for cracking. Actual deflection about 5 times "predicted" deflection.

If it is RC only, I would expect that deflection for short term uncracked. But 250 thick spanning 10m RC will be cracked even under construction load.

RAM Concept has several methods of estimating deflections. Hopefully you used the long term cracked version based on column/middle strips.

RE: Large Deflections in Concrete Slab Without Cracking

As rapt noted make sure your not looking at deflection plots for a standard load combination those are based on elastic first order analysis with gross sections so can wildly underestimate deflections. Set up some load history steps to follow the current life of the slab to get a better prediction of the deflection.

I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

Quote (Celt83)

Tough crowd.

You are correct. Sometimes we are too tough. But part of the trouble here is that we are given only half the picture and the picture that we are given isn't pretty. It would be good to know the boundary conditions (simply supported, continuous or edge) as well as if this is a PT slab.

The original post is suggesting a whole bunch of more obscure theories of why they are seeing the amount of deflection when they have failed to justify their calculated deflection which surely should be the first thing you do.

RE: Large Deflections in Concrete Slab Without Cracking

Ram concept had a large update somewhere between 2015 and now, the deflection results should be appropriate if modelled correctly.

In these types of situations I like to do the following steps.

1. Rebuild model in a program that is straight forward were I can control the inputs, rapt or strand 7 come to mind. The removal of props and no back propping will increase the creep deflection.

2. Change the model setup to reflect current day and current loadings. The increase loading until I hit the deflection recorded.

3. Check the weather, you mentioned water pooling. Find out what the weather was like if you don't live near by. Take these effects into account.

4. Map the deflection contours use a water level.

5. Figure out you percentage out from model to site. Get the stressing results if pt.

6. See if someone out side the construction company can confirm loadings.

7. Ask a colleague to review and confirm the above,

8. If nothing is still adding up, scan slab with radar and pull off live end caping. Check strands, reo layout etc, remodel from step 1.

9. If still an unknown issue I might get a second firm to review from scratch.

If you find there is a model stuff up. Don't worry this happens, call your insurance tell them findings, appoint a known expert for recommendations.



RE: Large Deflections in Concrete Slab Without Cracking

It seems like there are a lot of things to consider and examine before settling on crack investigation. I'd look at assumed vs actual end conditions first.

RE: Large Deflections in Concrete Slab Without Cracking

Another thing that hasn't been mentioned is differential shrinkage, which is likely to be significant if the top surface is exposed to full sun, but is probably ignored in the computer analysis.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: Large Deflections in Concrete Slab Without Cracking

What support conditions for this 10m slab? And what span length in the other direction?

As others have noted, L/d = 40 is quite a thin slab.

RE: Large Deflections in Concrete Slab Without Cracking

(OP)
Thank you all for the suggests. I'll be running through the recommended back checks and calculations throughout the day and will let you know the results. For now just wanted to provide some of the additional info that wasn't included in the original post.

The construction loading is 125/20
The design load is 134/40

The slab is a conventional reinforced flat plate slab w/ #6 at 12" O.C. T&B

Span is the middle span of a 3 span condition. The adjacent spans are both 21ft. the exterior supports are concrete walls while the interior supports are fixed columns.

This is not a transfer slab and the columns/wall above are not accounted for in the model of the lower slab.

The concept analysis used was based on the recommended 5 load step condition for ACI code requirements and based on the column/middle strips. When increase the construction duration to 180 days it increase the deflection to 0.75". The total estimated deflection after 5000 days was at 1.85". The external shrinkage restraint was set to 10% for the model.

RE: Large Deflections in Concrete Slab Without Cracking

"The external shrinkage restraint was set to 10% for the model."

This is likely too low if your exterior supports are concrete walls, under the new ECR settings that would typically land in the severe restraint category.

I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

3
I ran a simplified model based on your description:
10" thick Flat plate
21,33,21
Exterior Continuous Walls
Interior Columns (fixed at slab, pinned at foundation)
top and bottom mat of reinf. of #6 @ 12"

At 7-day load history with just self weight I get 1.03" of deflection:


I ran several more loads steps with 2 steps being 1 day of a +125 top surface temperature + Self weight and 1 day of a -125 top surface temperature + Self weight to cover a single cycle of + and - thermal exposure.
At 6 months load history I land on 2.66" of deflection:


I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

"...while the interior supports are fixed columns"

This assumption is likely no longer accurate as your column bars are probably set up to be continuous into the columns above and not developed into the slab. Since the columns up don't exist yet the bars aren't tied to anything and there is no additional compression from a floor above restraining the joint.

I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

I suspect that you need to take a much closer look at how your software is calculating deflection. Rapt is the one I would trust most on this subject....

I looked into slab deflection calculations for post tensioned slabs in great detail at one point. And, Rapt had some really good papers / descriptions on the subject. There are a number of ways to do it, some of which are described by code provisions. But, many of these are not really all that accurate.

RE: Large Deflections in Concrete Slab Without Cracking

Quote (OP)

while the interior supports are fixed columns.

What are the column details? How did you achieve fixity?

RE: Large Deflections in Concrete Slab Without Cracking

That span arrangement will change things.

We still do not know transverse span length.

Column stiffnesses will not have much effect with that span arrangement so no need to worry about how they work.

Is the reinforcement continuous top and bottom or confined to the tension areas on each face?

I assume 125/20 is LL and SDL.

RE: Large Deflections in Concrete Slab Without Cracking

I think 125/20 is DL/LL since 150 pcf times 10 inches is 125 psf and OP said the slab in 10 inches.

RE: Large Deflections in Concrete Slab Without Cracking

Quote (celt83)

At 6 months load history I land on 2.66" of deflection:

I just checked it and get similar. Around 65mm / 2.5" depending on configuration.

RE: Large Deflections in Concrete Slab Without Cracking

(OP)
The columns were detailed to not hook into the slab but similar to what rapt said, after modifying the fixity the deflections only slightly changed. In the transverse spans are 23',7',23' all supported on columns. GC Hopi is correct, it was 125DL/20LL.

In regards to the thermal loading I've never come across it until now but made a simple model similar to the image shown and was getting around 2.6 deflection as well but had it modeled as a 200' building. When I reduced it down to the building size I was getting around 0.8" deflection

RE: Large Deflections in Concrete Slab Without Cracking

Are you able/willing to share a screenshot of the actual floor plan, some things are getting lost in the piecemeal text descriptions?

I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

Rdar - In your original post you say that you were getting 0.6" deflection "under construction". I'm thinking that maybe this is where you are getting off track. When you say "under construction" that sounds to me like the day after the forms are stripped, i.e. with some construction loading, cracked but maybe not fully, and no long term effects. At this point, 6+ months out, you should basically be looking at total long term deflection - at 6 months you'll have a good portion of the lt effects baked in there and the slab weight is most of the total load anyway so I'd look there first.

In your analysis what was your total long term deflection? There's no way it would be 0.6" (that would imply L/660 for a 10" slab at 33ft, no way). 10" is very thin for a 33' span for any kind of typical deflection limits, at 33' even a 12" slab would be marginal. This is assuming 5ksi and minimal reinforcement as you stated (33' span with high strength/high mod concrete and jammed with rebar and you could do a bit better).

I'd take a step back - forget all the load histories, thermal, etc. and do some big picture thinking here. I'd say 1) Go grab any flat plate span/depth table such as this one, and you'll see that you're outside the typical recommendations. Granted those are ranges but that should be enough to at least make you suspicious that if you guys didn't do something special here then there's no reason to think your slab will outperform these ranges by much. 2) look at what your total long term deflection was supposed to be in your design. Not the "construction" step you are referring to, I mean total long term (5 yrs from now), does that jive with span/depth tables? It sounds to me like you're looking at cracked but immediate deflections, so expect a factor of say 2.5 to 3.5x that for long term, which at 6 months you are close enough to to not get bogged down in exact time history. The self weight of a 10" slab is the majority of the loading so expect to be near your total now.

What about strength? If you've got the capacity then this isn't the end of the world. You can probably muddy the waters enough by blaming delays, thermal etc. to have them level it (have to deal with that extra load), learn a lesson for next time, and move on.

RE: Large Deflections in Concrete Slab Without Cracking

For deflection, one item in your favor is that ACI requires L/480 for the portion of load applied AFTER attachment of the crack-prone elements....presumably drywall and finishes, in your case. It’s not total deflection. Meaning, one way to look at it is the clock hasn't started yet on your accrued problematic deflection (and your delay actually helps you more, letting creep accrue). Code also requires L/360 for live, which you will likely be fine for. That's it. Doesn't mean your slab isn't too thin, or that the 2.5" shouldn't be filled for basic serviceability and optics, but that's what the Code says.

RE: Large Deflections in Concrete Slab Without Cracking

Quote (Rdar43)

When I reduced it down to the building size I was getting around 0.8" deflection
Some far more experienced people than you or me have already done some quick calcs or models and have confirmed the 2.5" deflection. I'd stop persisting for the time being with you current approach and try another. I suggest you try some hand calculations or referring to span tables.

I'm a mostly a steel guy so I'm fairly inexperienced in calculating long term deflection of concrete slabs. But I can quite readily look up tables like bookowski, my favourite is Guide to Long-Span Concrete Floors It is easy to see your slab lies well outside of normal ranges.

(I'm sticking around in this thread because I'm aware that I DO need to learn more about long term deflection of concrete!)

RE: Large Deflections in Concrete Slab Without Cracking

Quote (Rdapts)

around 2.6 deflection as well but had it modeled as a 200' building. When I reduced it down to the building size I was getting around 0.8" deflection

Post a sketch of the floor plan so we can understand what you’re talking about.

RE: Large Deflections in Concrete Slab Without Cracking

(OP)
Below is a simplified floor plan of the slab

RE: Large Deflections in Concrete Slab Without Cracking

With that arrangement of perimeter concrete walls your shrinkage restraint should definitely be set to severe.

I’d still expect the long term defl to be in the neighborhood of 2”+.

I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

I’m not sure how that arrangement improves things from 2.5” to less than 1”

RE: Large Deflections in Concrete Slab Without Cracking

I land on 2" after load history considering temperature effects:
From the manual load history does not consider load patterning so the 20 psf construction live load would be considered over the whole slab, if they stacked material in the central bay only that would increase the deflection. Concepts method iteratively cracks the design cross sections if your number of design cross section divisions is too large not enough areas of increase curvature will be picked up it then averages these stiffness over each finite element, the analysis is also still linear elastic so any p-delta effects aren't captured. The mesh size itself will also impact the results a larger mesh will be inherently stiffer than a finer mesh.


I get 0.8" with just self-weight after a single 10 day load history (3ft mesh, 2.5 ft max section division, per the manual though results less than 90 days contain a lot of variability not captured by the analysis so the validity of the results is questionable):



Long term (250 days +/-, 10 days initial, 60-days of construction activity, 180-days of inactivity) with a 1 ft mesh and 1 ft max spacing for the latitude strip divisions, 2.27" total deflection:
again this wouldn't capture any patterning of the construction load. the 3' mesh size was a good estimate, this tighter mesh is an even better estimate but we still haven't captured everything that impacts the deflection.



I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

What potential problems have been brought up with the current 2.5” over 33 ft? What type of partition walls will be on the slab etc.. And are you confident in the strength of the slab?

RE: Large Deflections in Concrete Slab Without Cracking

Quote (Rdap13)

made a simple model similar to the image shown and was getting around 2.6 deflection as well but had it modeled as a 200' building. When I reduced it down to the building size I was getting around 0.8" deflection

Did you make any other changes in the model? As Celt83's results indicate, the reduction in transverse span only slightly improves the overall deflection. It doesn't make sense that deflection would reduce from 2.6" to 0.8" by reducing the transverse span.

RE: Large Deflections in Concrete Slab Without Cracking

Rdapt13 - I have the same question as Tomfl. Assuming that the span in question is an interior span, why would building length (in either the longitudinal or transverse direction) significantly affect slab deflection? (i.e., 2.6" versus 0.80"?)

RE: Large Deflections in Concrete Slab Without Cracking

Did you have the construction team pressure wash the slab or anything to look for cracks? I would expect cracks in this slab design shortly after the forms were removed and by now I do not see how they would not be visible. The slab image posted is basically symmetrical about the vertical line, so I modeled 1/2 the slab with some arbitrary supports. The image below show the estimated cracking with only the dead loads.



I am not sure where the OP is located, but based on the 20LL, it seems it is not NA/AU/NZ. In NA I would expect a Peer review would start now. I am not sure how much more the OP will share. I am curious how others might approach a repair for this assuming they were engaged to look at this.

Ideas that come to mind:
1. Demolish and start over.
2. Hire Ingenuity to come up with a PT repair.
3. New beams??
4. Do nothing.

RE: Large Deflections in Concrete Slab Without Cracking

Those are fairly expensive solutions to a deflection issue, assuming there is a deflection issue, while the eng-tips feels like it is too much, Generally deflection issues are engineering judgement on use of floor, if it was a carpark use would we do anything? If it was offices would we do anything? Deflections are normally around finishes and looks. deflections don't constitute a failure and unless the client had a specific contract around deflections, a long argument is ahead as to what is a failure.

There were a number of buildings in the early 1990's that had deflections in the order if 3 inches. In most of these cases the engineer had enough strength for the finishes to take up the deflection, with a self leveling grout or similar.

RE: Large Deflections in Concrete Slab Without Cracking

2 1/2" deflection during construction is terrible and exceeds l/180. I am curious what Celt predicts the total long term creep would be. Adding mass would level the surface, but it will add to the problem as well. If one is going to play with span/depth like this one must be far more careful.

RE: Large Deflections in Concrete Slab Without Cracking

It’s an interesting discussion as to whether it really is a problem, but OP’s question wasn’t “is this a problem?”, it was “why do I have this problem?”.

RE: Large Deflections in Concrete Slab Without Cracking

Tom, yes, that is true. This is the point in the discussion where the poster usually goes silent since they did not get the answer they were hoping for. I suppose I could start another thread to discuss.

RE: Large Deflections in Concrete Slab Without Cracking

(OP)
I apologize for not responding, been on vacation for the past week and return tomorrow. I'll try and find some time to go through the comments and address them

RE: Large Deflections in Concrete Slab Without Cracking

(OP)
To answer the question I can right now:

1) This is the 1st time I've considered temperature during construction as an effect on the deflection of a slab so there is a good chance the load steps I used were inaccurate to account for the difference between 2.6" and 0.8". I agree that there wouldn't be much of an effect for a continuous slab but, my initial thought process was the larger the slab, the more thermal expansion would occur and thus would potentially produce more cracking and stresses. For those who ran a model, if you could provide the exact load steps you used to get a larger deflection for me to confirm with my model that would be greatly appreciated.

2) The building is located in New Jersey, without a location a temperature range from -125 to 125 makes sense but over the period of the slab construction (March - September) I've estimated a temperature between 40 to 125 would be accurate.

3) The 0.8" mentioned previously was for a 6 month load step with 3ft mesh. Long term deflection (5000 days) the slab was right around 2" deflection with moderate ECR in concept, changing it to severe would likely bring it to the 2.25 range. I'll run it at 1ft mesh when I can. I imagine this will increase the deflection but would be naively surprised if reducing the mesh to 1ft would cause such a large increase in deflection. The lack of validity for concept model less than 90 days is an interesting consideration as I imagine this slab has been through the worst case conditions.

4) Ownership has gone with a similar route mentions by bookowski. The strength of the slab still works, pin the deflection on the pause and learn from the span/depth ration error that occurred. Regarding haynewp's comment, I don't think there has been an issue yet. The plan is to pour a topping to make any partitions above level and below is intended to be an open area.

5) For the span/depth ratio concerns, my interpretation (and those of my coworkers) is that there is no indication of loading for these ratios. So the slab could be loaded with a significant amount of load, let say 35SDL/125LL, and the same span to depth ratio would be used for this project 15SDL/40LL. Should the same span/depth ratio's be followed or should there be an increase due to the reduced loading?

6) Brad805, we haven't had them pressured wash it but we had them do a survey to estimate the additional concrete topping required so it should be fully cleared for another indeth inspection. The large deflections and lack of cracking was what led us to believe that there was a construction issue rather than a design issue to begin with but after reading these comments perhaps that isn't the case. The increased topping will increase the demand but if it causes strength failure we are planning on adding additional supports/reinforcement.

RE: Large Deflections in Concrete Slab Without Cracking

Here is the load history I used:
All Dead = self weight of the slab only, I did not apply any superimposed dead loads
CL = L = 20 psf on the entire slab for Construction Live Load
Tp = Thermal load of +120 top +40 bottom assumes sun exposure on the top surface
Tm = Thermal load of -40 top and -40 bottom assumed uniform cooling


I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

Temperature and live load differences won’t explain the difference. OP gets 0.8”, and other people (and the slab itself) get 2”+.

RE: Large Deflections in Concrete Slab Without Cracking

If op ran a single load history step of 180 days, that would result in a different result as the creep and shrinkage computations would be based on 180 day value and the slab would start uncracked. I've found you really need to break it down into the "load history" over those first several months and then a longer sustained period, this tends to result in much more early life curvature that then propagates through the later histories.

I'm making a thing: www.thestructuraltoolbox.com
(It's no Kootware and it will probably break but it's alive!)

RE: Large Deflections in Concrete Slab Without Cracking

I have been noodling this problem for my own continuing professional development off and on for a bit. I thought I would share a few thoughts.

I used loads similar to Celt. Mapping the time steps in the software I use is a tad bit more onerous, so I created a spreadsheet of the functions needed.



The first analysis is below. The deflection in this case was much lower than reported. This analysis assumed the concrete age, t0, was 28d when the load steps started. That was not the case here.



I re-ran the analysis using a t0=5d. Here I did not include the temperature load case. The deflections are closer to what is being reported.


Now you may say, but we used a high early admixture. I think you have created a design that is very sensitive to creep and shrinkage. Do you know how the curing methods or admixtures affect the hydration process? You need to look at that carefully. Below are some excerpts from the CEB fib Model code 2010. In this you can see how the age of loading affects your creep parameters. Shrinkage can be affected similarly. I am sure Celt adjusted his parameters correctly, but I am not sure you have nor considered this in detail.



To show the impact of the age of loading on the basic creep coefficient I created a simple calculation of the value for 3/7/28 d below.


Some general recommendations of the steps needed if you have a design sensitive to creep and shrinkage are below. This is a from a 1973 study, so does not have current admixtures. I did not find any other free documents that might include current practices, but I did find a number you could pay for.


At this point you probably think, but everyone uses high early admixtures and strips forms as we did. The key difference is most do not use designs with this span/depth ratio. To show the difference I reran the design if the slab were 15" thick. I am not suggesting that is an appropriate design nor cost effective. It is merely to illustrate the benefit of using the span/depth ratios suggested. This analysis was subjected to the same load case, and the t0 = 5d.


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