1950's two way reinforced concrete slab analysis 3

[VESE]

I've come across a reinforced concrete building with a two-way slab built in 1957. The slab thickness is a constant 10" (no edge members, no column collars) the slab live load is 300psf, the slab compressive strength is 4,000 psi and the reinforcing is ASTM A15 Intermediate Grade (40,000 psi). When I look at the current ACI Direct Design method, the slab reinforcing is over stressed in negative moment by roughly 50%.

The Direct Design method is based upon uniform loading, which seems odd because I would think it should be based upon a triangular loading. The bays are equal, so this means a rather large reduction in bending moment - to where it meets the slab capacity or under a little under even.

So my question is - do you know of any commonly used method that would allow for the triangular loading? Why is the uniform loading procedure used in ACI? Like I said - that method seems extremely conservative.

It's always possible it's just a mistake and it's never been picked up because of redundancy and/ or never utilizing the posted live load capacity.

Thanks,

 

[BAretired]

Triangular loading? Why would you think that? It is usual to base the design on uniform loading, but in the case of a live load of 300 psf, it would be usual to consider unbalanced loading, i.e. checkerboard or skip loading.

BA

 

[VESE]

Since I know the original loading, material strengths, geometry, and size of bars, and the fact that this floor has been in service for almost 60 years, my guess is that there is some way that this system works with little over stress. I'd like to believe that the original designer didn't make such a large mistake.

I understand the pattern loading, but neglecting that for a second, designing the slab to act in each direction under full load seems very conservative. Since the aspect ratio is one, the load distribution from a uniform area load would produce a triangular loading for the floor plate. Or at least it would if I was looking at a steel plate floor. For concrete, I would follow whatever design method/guide was available. Hence the Direct Design method.

Now the pattern live load would produce a larger positive bending, but my problem is more on the negative side. Oddly enough, the positive bending regions seem plenty adequate. So I was just wondering if this triangular loading or some alternate load distribution was ever used as a basis to reduce the negative bending moments the slab would have to take.

 

[BAretired]

Going back to the origins of the flat plate, a guy by the name of C.A.P. Turner patented the system a little after the turn of the century, I think it was about 1905 or so. His system involved reinforcement in four directions. A Professor Eady produced calculations which nobody could understand purporting to show that flat slabs could be reinforced well below the statical moment requirement. Amazingly, the system seemed to work just fine but eventually a guy by the name of J.R. Nichols demonstrated in a very simple way that two way slabs had to carry the full moment in both directions.

Perhaps it seems very conservative to design the slab for full load in each direction, but it is not. It is the way the system works from a theoretical point of view. Nevertheless, notwithstanding the outstanding and well recognized paper by J.R. Nichols, for many years the code required two way slabs to be designed for moments less that the statical moment which should have prevailed. ACI 318-63 required that the numerical sum of the positive and negative moments in a flat slab be not less than Mo = 0.10WLF(1 -2c/3L)[sup]2[/sup] where F = 1.15-c/L but not less than 1. I don't have in my possession any earlier codes, but prior to the 1963 code, the sum of positive and negative moments was only 72% of the statical requirement if my memory serves me correctly.

More recently, both the Canadian and American codes have been revised to satisfy statics, but it must be admitted that over time, two way slabs have demonstrated a resilience which cannot really be explained by theory.

The distribution between positive and negative moments in two way slabs is another matter. Codes have varied substantially on that point.

If the numerical sum of the positive and negative moment resistance is about 70% of the simple span moment in your case, the slabs were likely designed in accordance with the code of the day.

BA

 

[paddingtongreen]

I know there were several different rationales, I don't remember them too well as I was working structural steel myself, I just sat with some of the guys doing concrete.
I do have questions:
What is the column grid?
Is the reinforcing evenly spread or in a pattern?

There was a "triangular" load pattern in one of the rationales, but I think that it may have increased the negative moments! I'll have to give it some thought.

Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin

 

[VESE]

Thank you very much! That was exactly what I was looking for. I knew the methodology has changed from ASD to LRFD since then, but I wouldn't have thought that would make that much of a difference anyway. But the change in design methodology was exactly what I was hoping to find.[pre][/pre]

The total design moment is roughly 80% of the simple span design moment using a uniform design load, which sounds like it would've been acceptable based upon the standards of the day.

 

[dhengr]

BA:
By triangular loading, does he mean... draw lines, in plan, btwn. the four columns carrying a panel, not on the column lines/grid lines, but diagonally. These diagonals show four triangular load patches which go to (load) each column strip? Otherwise, nice history, I would not have come up with those names and history off the top of my head. Happy 4th, even if you have a little less reason to celebrate that date.

The reason that old slab works is that the 40ksi steel probably had a higher yield than the min. reqr’d.; and the same for the 4ksi conc. Then, those slabs are seldom loaded to the design cap’y., and there is also a fair amount of redistribution of loads and moments as the entire slab works under load. You have a better handle on this topic than I do, but wouldn’t yield line theory bear this out?

 

[VESE]

That is what I meant by triangular loading. Basically, your slab load varies from 0 at each column line to a maximum of q*L midway between the columns. This loading is half of what the current Direct Design method, and I figured that there was a reason why it isn't done this way now, bit wasn't sure if that was based upon newish research or older research.

 

[VESE]

PaddingtoGreen - the bays are 20'x 20' and the reinforcing is broken down into column strips and middle strips in each direction. The only thing that is a bit off is the fact that the negative moment capacities are 66% of what they should be.

 

[rapt]

Nice history BA. A lot of designers in the years before the 50's apparently designed for half of the load in each direction. Presumably they deflected sufficiently for catenary action to help!

Vese,

But then the column strip has to get the load dropped onto it from the triangle to the columns. And the total carried in each direction is eventually 100%. FEM analysis including Msy moments will give exactly the same result. This is assuming an orthogonal reinforcing pattern. If you want to reinforce for principal moments, the requirements are reduced, but the angle of the moments is different at every point in the bay, so this is impossible with reinforced concrete. It would work for an elastic material.

The triangular logic only works if you have a continuous support on the sides of the bays, not point supports like columns!

 

[paddingtongreen]

As I remember it, the diagonals only went from corner to corner in square or nearly square grids, in elongated grids the lines would be run in at 45[sup]o[/sup] till they met in the middle and the two new points connected; it would look much like a yield line diagram for an elongated panel.

I, and the people I worked with back then, looked at the continuity through the column strip, and provided adequate (at the time) negative reinforcing.

@rapt. Don't knock the old guys, they didn't have the facilities or the research that you can access and yet they built structures that last and didn't have a higher failure rate than I see today. They did know that there was much they didn't know and worked around it without ignoring it. I am wondering where the catenary comment came from.

The triangular logic relies on equal deflections of the parallel column strips but assumes the same deflected shape across the building but different amounts increasing from col. line to halfway and decreasing up to the next col. line. The outer strips provided only simple support.

Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin

 

[BAretired]

dh and paddington,

You understood the OP correctly. By triangular loading, he meant a triangular area in plan as bordered by the diagonals. This is wrong in principle as rapt pointed out. It would be correct for a two way slab supported on solid walls all around but it is not correct when the supports are columns.

It has been a number of years since I read J.R. Nichols' paper on the subject, but the principle is clear and I will never forget it. Essentially what he did was cut a section through the slab in two orthogonal directions at the center of the columns. He showed that for any bay, the sum of positive and negative moments in each direction had to be wL[sup]2[/sup]/8. The reinforcement in one direction did not benefit from the reinforcement in the orthogonal direction. This principle is very clear when one thinks about it and it is somewhat surprising that it took the engineering community so long to realize and correct the code to accommodate it.

BA

 

[BAretired]

Found it!

Portions of ACI318-56 were incorporated in "Reinforced Concrete Fundamentals" by Phil M. Ferguson, copyright 1958 by J. Wiley & sons. The attached link is an excerpt from Article 1004-"Design by empirical method" of the 1956 ACI code which shows Mo = 0.09 WLF(1-2c/3L)[sup]2[/sup] for the numerical sum of positive and negative bending moments. So in 1956, engineers who chose to use the empirical method were designing for 72% of the statical moment.

BA

 

[Tomfh]

I had this argument with a fresh graduate a month ago. He was arguing that there's no way you need full reinforcement in both directions. he said you only needs 50% each way. 50% + 50% = 100% of the load, QED. We almost convinced him otherwise, but he didn't really beleive us. He was like "look, if the code says I have to, I will, and at the very least it will be conservative". lol.

I've never really understood the confusion myself. And I'm a bit surprised you need a paper or theories of static equilibrium to prove it. It's entirely self-evident!: In a one-way slab you obviosuly design for the full load in one direction. How on earth can deleting 95% of each supporting wall reduce the bending load in the original direction?

 

[Tomfh]

I have a 1918 Hool and Johnstone concrete design text with a lot of intersting stuff about flat slab history. I don't know how it was in the 50s, but this 1918 background gives an idea of where things developed from

In 1918 when the book was written msot flat slab designs were specialist patented systems. These slabs have astoundingly complex reinforcement by todays standards. Four-way systems (i.e. there is diagonal reo between columns. "three way" Hexagonal patterns. "Spider-web" patterns. circular patterns. Here is one such example, the S M I flat slab. The dark lines are the radial reinforcing, the fainter lines the circumferiential reinforcing.

These crazy systems were apparantly based on real load testing. They'd load up with twice the actual service load for over a year.

Now, there is also "the chicago ruling" of this era, which is a theoretical alternative to load tested proprietary systems. This chicago ruling is essentially wL^/8 apportioned to column strip/middle strip, as most of us design to today. The text refers this approach as the "best and most conservative".

The book comments that "surprisingly low stresses have been measured in the steel for slabs designed as per the Chicago ruling". The stresses are in the order of 1/3 to 1/4 what calculation would predict. So maybe the graduate whose opinion I just dismissed might have a keener insight into slab behaviour than I gave him credit for!

 

[msquared48]

The 1963 ACI code, portions of which were included in my Ferguson from 1966, were distribution ratios for moments, positive and negative, for several different edge conditions and geometries.

Mike McCann, PE, SE

 

[VESE]

Thank you all for your responses! And I'm extremely grateful for the invaluable history of the code development in this area.

I also found an older design guide referencing the 1963 ACI provisions from our company's library as well as some of the earlier provisions. It doesn't reference it directly from any specific paper or code, but it does note the Mo = 0.09WLF(1-2c/3L)^2 equation and then the moment distribution beyond that. This was the answer I was hoping to find.

I don't actually do a lot of concrete slab design and I've never designed a two-way slab, so my impression of what seems conservative is based upon steel construction and I guess I was thinking of what the loading of a beam between the columns would be, which is not at all my case. Point well taken.

But at any rate, thank you again. I owe you each a beer.