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Here's a nice little mid-century house near Detroit, in remarkably original condition. In this photo note the cantilevered stair landing; three holes that I'll assume contain post-tensioning points are placed above the centers of the trio of little slabs.
https://www.zillow.com/homedetails/4980 ... /?mmlb=g,2
The things I thought were seared permanently into my brain had dissolved and recomposed into fictions of memory.
Let this be a lesson. Do thou NOT as I have done.
So from the book:
First, a moment is a force in rotation about a certain point.
Example, say there is a 75Lb point load on a simple beam 5Ft away from the left hand support A.
If the center of rotation is taken at point A then the moment is the product of the force and the distance:
75lb X 5Ft = 375 Lb-Ft.
Second, that external moment load (375 lb-ft) must be resisted inside the beam with an equal force in the opposite direction.
This resisting force or stress is called the internal bending moment (M).
Third, (just in terms of bending, not shear or torsion) forces will bend beams in one of two ways (or in a combination of the two depending upon loading and support conditions).
The first way is to be bent down into a 'smile'. This is the shape of a 'positive internal bending moment'.
The second is bent up into a 'frown'. This is 'negative internal bending moment.
That's enough for now.
I don't have a guess yet exactly what Glickman is referring to by "negative reinforcement." He could be using it idiomatically. In any case, whatever the direction the section of the beam was bending - he didn't put it there.
Finally, the bending I've been talking about at FW is that in the E/W direction along the main level balcony. More to come on that.
Yet JimM, seems to say that at least some critical cracking was in the bolsters.
The files do have one drawing of the reinforcing in the bolsters. I'll get to that here eventually.
As Tom explained, there is a point where forces (moments) will tend to bend-or rotate-a beam (bending moment is actually a property occurring within the beam). While bending moment occurs at some point along a beam supported at both ends, a cantilevered beam fixed at one end has negative moment along it's entire length. Unlike wood or steel beams, where material properties dictate structural capabilities (at a given span), the size, quantity, and precise placement of reinforcement in a concrete beam determine how stresses are controlled. Generally, the upper half of a beam is subject to tensile stress, where a configuration of reinforcement is placed. The lower half is subject to compressive stress, where likewise reinforcement would be placed. FW was continuing to deflect even at the time post-tensioning was performed. The steel cables when attached to the beams more or less emulated where tensile and compressive correction was required, bringing the cantilever into the equilibrium it never had, if artificially.
Yes SDR, I did not intend to infer the bolsters cracked... I don't think so, anyway. Simply put, deflection was caused due to inadequate reinforcement placement, including a system which would have included the bolsters. Without it, building loads were transferred to the bolster end of the beams at which point they acted as shear forces, causing the beams to crack.
---rotate downward. In doing so, as a continuous and homogenous member it tries to remain straight, so if the unsupported end rotates downward, the beam as a whole curves more or less continuously, meaning the portion inboard of the fulcrum would actually rise.
I'm thinking here of a steel beam. A reinforced concrete beam, to act in the same way, would want to have its rebar placed both low and high---to produce a beam that would act like a steel shape, in cantilever. Then, to keep this cantilevered concrete beam from dropping at one end and rising in the inboard segment, most of the reinforcing ---which resists tension---would want to be placed near the top edge of the beam ?
The post-tensioned repair, on the other hand, placed the tensile members in a shallow inverted V---low at the inboard end and at the free end, rising to the highest fulcrum at the middle for which vertical clearance could be obtained. . .
... I've got a sketch but trying to remember how to get it here. GDorn, can you look this over and let me know what I need to delete so the image will show? This is the entire codeline, I don't know why some is red and some is black. Also this may not be the right codeline. I pasted it from embed tag in the Flickr share function.<a data-flickr-embed="true" href="https://www.flickr.com/photos/54449844@ ... ateposted/" title="FW:WestTerrace"><img src="https://live.staticflickr.com/65535/500 ... 7ba7_o.jpg" width="2200" height="1700" alt="FW:WestTerrace"></a><script async src="//embedr.flickr.com/assets/client-code.js" charset="utf-8"></script>
In the meantime here it is on Flickr: https://www.flickr.com/photos/54449844@ ... otostream/
1" dia. rod clear around parapet.
1" dia. rod w/ 90 degree 12" bend at each end in cantilever joists on west and east sides of balcony.
Same rod at same height in north parapet on the end.
Eight - 1" sq. rods at all beams over bolsters.
Extending 7'-0" from front of Bolster back into beam, no hooks either end.
Notch joists 12" wide X 9" deep entire length of beam between 3 and 4.
Notch at 3 joists between 2 and 3.
1/2" dia. rods 9" o.c. in entire bottoem of slab bent up into parapet at ends.
Cross reinforced w/ 1/4" dia rods @ 12" o.c. bent up into front parapet same as 1/2" dia. rods.
Place two 1/2" dia rods 2" o.c. under all joists.
1/4" dia. horiz. rods 8" o.c. clear around parapet wired inside of vert. rods - same in stair parapet.
Reinforcement @ corners of parapets 1/2" dia. rods extend 36" each way placed between 1/4" dia horiz. rods.
Anchor 1" dia. rod in top of parapet into stone pier
Apologies for my lack of thoroughness, something was still not clear enough after mainly taking a stab at what a cantilever "is". Having said that, and it may just be me getting my bearings straight, but in fact, it was the upper terrace which could not act as a cantilever due to Glickman's error. Also, Sillman included the structural piers stacked above each other on both levels as part of the "bolster", which are located at the end edge of each lower bolster, and is where cracking did occur on both levels. There is reinforcement in the upper part of the concrete beams of the terrace, but it stops at the piers and should have continued through the beam to the south parapet. These beams were an important part of the structure as cantilevers, and the window frames intended to only transfer only some of the load. Without that huge terrace load primarily being supported by cantilever as intended, the entire load was placed onto the main floor beams.
Even though it was placed correctly, as shown in Tom's drawing, the additional reinforcing in the lower beams was still not enough and surely not enough to handle the weight eventually imposed. The steel that was there was found to be at 95% of it's yield strength. Silman also pointed out that it was also very fortunate that the concrete at FW was very good, 5000 psi, and creeping (shortening/compressing over time) normally yet at 4500psi working capacity. BTW, they also post-tensioned East/West end to end, which helped relieve stress of the balcony's on the beams. SIlman believes what was done at FW is evidence that Wes and Glickman most likely approached the solutions correctly, but over the course of time and most likely innumerable distractions and revisions, simply forgot to return to the requirements of the upper terrace. The steel suppliers were also engineers, and along with Kaufman's engineers, advised there was not enough steel in the main floor beams.... luckily there was-for a while!
It's amazing, isn't it, that this omission should have gone unquestioned at the time of construction, by those doing the work ?
I'd forgotten all about that.
For some reason I'm attracted to how that second level terrace is structured so this gives added incentive to investigate.
I'm going to draw that out as far as I can as well.
No competition in the drawing department for SDR of course but at least the information will be conveyed.
Did you figure out why your images are not posting here ? I tried posting one of them myself, using the provided protocol (that little "landscape" icon/button), but no dice. Perhaps not all flickr accounts provide web hosting ?
You can always send me items to post . . .
Jim's contributions are always most useful. I've forgotten what Silman said about the interaction between the first and second-floor cantilevers, via the steel mullion frame. Ideally, I suppose, there would be neither tension nor compression on the vertical members of that frame---resulting from equally-effective cantilevering of both levels ?
"... Edgar Kaufmann Sr., who commissioned "Falling Water," one of the world's most celebrated modern houses, destroyed all his Wright material before his death."
Can anybody confirm that? I've never heard of this before (even though I've had this book for decades).