Because my degree is in naval architecture and marine engineering, things that float interest me. Though it may seem incongruous, this nautical background actually has come in quite handy when it comes to envisioning slab-on-grade behavior, because essentially:
Rule 11a: Slabs on grade are just very shallow blocks of concrete floating in pools of very dense liquid.
This “floating block” analogy is easily accurate enough for everyday work and leads directly to Archimedes' Principal for slabs on grade:
Rule 11b: The displaced grade volume multiplied by the grade's equivalent liquid density equals the weight of the slab.
The grade's equivalent liquid density—measured in pounds per cubic inch (pci)—is an imaginary property called its Westergaard Sub-grade Modulus or k value. If the grade were a liquid, then the liquid would have to have a density of k pci in order to provide the same buoyancy to the floating slab as the actual grade material appears to do. Rule 11b says a t-inch-thick slab made of p pounds per cubic foot (pcf) concrete must sink p x t / 1728 k inches into the grade. Because k values usually range between 50 and 250 pci, immediately upon placement, every new 6-inch slab typically will sink somewhere between 1/100 inch (the thickness of a playing card) and 1/500 inch (the thickness of a sheet of paper) into the grade.
If new slabs would only stay put, then calculating the stresses induced in them by their later live loads would be a fairly straightforward exercise. But they never stay put. As soon as the concrete is placed, the exposed tops start to dry, shrink, and curl. Then their bottoms routinely morph from uniformly supported static planes into contracting spherically shaped dishes that lose ever more support around their edges as they continue to curl up and sink ever deeper into the grade.
Assume that a plain t-inch-thick slab panel made of p pcf concrete is L feet long and W feet wide, so that its weight w = p x t x L x W / 12 pounds. Now let this slab curl up without cracking into a spherically shaped dish, and let s be the depth in inches that sinks into the grade (see diagram). The volume v in cubic inches of the displaced grade material then will equal about s2 x 107 / C, where C is the slab's Curl Number (See “Curl Numbers” in the August 2010 issue of CC. Note, however, that the numerator in the Curl Number equation should be 10,608,000 S, not 6630 S). Because Rule 11b states that v x k must always equal w, a little algebra shows that s2 must equal w x C x 10-7 / k, so that:
Rule 11c: On akpci grade, the centroid of a plain, crack-free slab weighingwpounds and having a Curl Number ofCwill sinks= [w x C/k] ½ / 3162 inches into the grade.
Suppose our assumed slab is t = 6 inches thick, is made of p = 150 pcf concrete, is placed on a k = 150 pci grade, and is sawn into 15x15-foot panels. Because each panel weighs 16,875 pounds, Rule 11b says that before they curl (while the Curl Numbers are still zero), all the panels must sink uniformly into the grade about p x t / 1728 k = 150 (6) / 1728 (150) = 0.003 inch.
Now assume the panels remain crack free for two months, but curl enough to raise their average Curl Number to 50. Rule 11c says that the centroid of each curled panel must now have sunk [16875 x 50 / 150]½ / 3162 = 0.024 inch into the grade. Solely as a consequence of the curling, therefore, the pressure imposed by the concrete's deadweight has been concentrated toward each panel's centroid enough to cause them all to sink another 0.021 inch into the grade.
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