Even within Civil Engineering, this stuff is fragmented.

Wood, Steel, and Reinforced Concrete all behave very differently and are studied separately in a structural context. This is a pavement problem, which is a totally different animal because the structure (slab) is continuously supported by the ground. Simple statics don't apply and a lot of geotechnical voodoo gets introduced.
You don't have strength issues to worry about if you follow the slab specs from the manufacturer. The key thing you'll want to realize with your slab is that the rebar is not structural. In your slab-on-grade, its there (as one tool) to help you control cracking as the slab cures. If you want to get deeper in to the engineering of this stuff, the "ACI 360R Design of Slabs-on-Ground" by the American Concrete Institute is the place to go.
Your big complication (and why the dogbone will crack the slab) is that the cured concrete material is very different from the form it starts out as it leaves the concrete truck. As it undergoes its chemical reaction and cures, the material shrinks.
In general - If the slab were poured on a smooth sheet of glass - no problem. It would just shrink away from the four walls and be smaller than when it was first poured. However, in reality, the slab is interlocked with the base material and this friction acts to rip the slab apart as it cures. As a result, tensile stresses build up in the slab. Concrete is weak in tension and it subsequently cracks. So, we place saw cut joints every 10-12 feet in the slab so weak points are introduced. Cracks then form along these "control joints" so the end result isn't so unsightly. (Cracks can still form outside these lines - its all a game of probabilities.) When rebar is placed in the slab, it helps resist these tensile forces and allows one to place the saw cut/control joints further apart - or it can be used as extra insurance against cracks forming outside the control joints. (In some cases, with enough steel, the joints can be eliminated entirely.)
When you have a piece of steel, (like the dogbone) a penetration, or other foreign object embedded in the slab it is immobile and creates a discontinuity in the cross section of the concrete. (just as the saw cut joint does) So, as the slab cures, it divides at these points and "pulls" away from these areas, forming cracks that radiate out from them.
In situations where penetrations, pipes, steel columns, etc. can't be avoided in slabs, the saw cut/control joints are planned so that they intersect these objects.