Whats the relation between the centre of gravity of a structure and the middle third rule?
If you elevated the entire structure and determined where the fulcrum point should be, which would behave as a balanced teeter-totter with equal weight on both ends, you will find the center of gravity. If one end of the structure is heavier than the other or if the foundation is of softer material or sloped, then the structure would be likely to tilt toward the heavier end, end with softer material, lower end of the slope, unless you shifted the fulcrum (center of gravity) closer to that end.
Middle Third Rule suggest that if the fulcrum point is in the center, then gravitational stress would be trivial in the center third and critical at the outermost point of each end. Consider the Teeter Totter above; applying a 1 ounce weight on both sides of the teeter, near the center, the teeter is not likely to tip one way or the other. Placing the weights on one side near the center and the on the outermost edge of the other end, will cause the teeter to tilt dramatically toward that side.
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